Steps which lead me to more of an understanding:

1. Understand Rotation vs Reflection

1. The Orthogonal Group $O(n)$ contains all distance preserving transformations of euclidean space. (determinant equal $\pm 1$)
2. There are two connected components (sets are connected, if they can not be the union of two open sets), i.e. the two components are both connected sets but disjoints, so $O(n)$ itself is not connected - there is a discontinuity.
3. The special orthogonal group $SO(n)$ is the connected component containing the identity. It is a normal subgroup and we also refer to these operations as rotation. Its elements have determinant 1.
4. The other connected component are the reflections and have determinant $-1$. Since two reflections have determinant $1$, they do not form a group.

Since the reflections are not connected to the identity it is not possible to continuously transform an object into its reflection. Objects which are different from their reflections (even modulo rotation - continuous distance preserving operations) are called chiral.

2. Understand that determinants measure a (signed) change in volume.

Where the sign indicates whether a reflection took place. The sign is enough, because two reflections are simply a rotation no matter the dimension.

Proofsketch: Let $A$ be a linear map. $\lambda(A \cdot)$ is still a translation invariant measure, so by uniqueness of the Lebesgue measure (as the only translation invariant measure up to constants) it is just the Lebesgue measure up to a constant. To calculate the constant it is sufficient to consider $\lambda( A [0,1]^d)$. For diagonal $A$ it is immediately obvious that it should be the product of entries. But with the singular value decomposition we can decompose any general matrix $A$ into a positive definite diagonal matrix and distance preserving members of the orthogonal group.

3. Understand the set of alternating multilinear maps

Since the set of alternating $n$-multilinear maps in $n$ dimensions is one dimensional, the determinant is its basis with the property that the determinant of the identity is $1$.

Now we understand multilinear maps as a (weighted) measure of volume change. If we were to measure $k$-volume in $n$-dimensions, we have more possible directions in which we can measure volume, so we stop having only one basis element (the determinant) and instead get multiple. But the principle remains the same. :handwaving: (maybe someone can one up me here)

Steps which lead me to more of an understanding:

1. Understand Rotation vs Reflection

1. The Orthogonal Group $O(n)$ contains all distance preserving transformations of euclidean space. (determinant equal $\pm 1$)
2. There are two connected components (sets are connected, if they can not be the union of two open sets), i.e. the two components are both connected sets but disjoints, so $O(n)$ itself is not connected - there is a discontinuity.
3. The special orthogonal group $SO(n)$ is the connected component containing the identity. It is a normal subgroup and we also refer to these operations as rotation. Its elements have determinant 1.
4. The other connected component are the reflections and have determinant $-1$. Since two reflections have determinant $1$, they do not form a group.

Since the reflections are not connected to the identity it is not possible to continuously transform an object into its reflection. Objects which are different from their reflections (even modulo rotation - continuous distance preserving operations) are called chiral.

image source

chirality allows us to understand why the multilinear maps we will consider later on should be alternating. I.e. swapping two coordinates flips us into the reflections. And swapping two coordinates will get us back to rotation. If one of the coordinates you swap is the same, then this operation only touches 3 dimension, so even in higher dimension it is possible to visualize how this is only a rotation since the other dimensions remain untouched.

2. Understand that determinants measure a (signed) change in volume.

Where the sign indicates whether a reflection took place. The sign is enough, because two reflections are simply a rotation no matter the dimension.

Proofsketch: Let $A$ be a linear map. $\lambda(A \cdot)$ is still a translation invariant measure, so by uniqueness of the Lebesgue measure (as the only translation invariant measure up to constants) it is just the Lebesgue measure up to a constant. To calculate the constant it is sufficient to consider $\lambda( A [0,1]^d)$. For diagonal $A$ it is immediately obvious that it should be the product of entries. But with the singular value decomposition we can decompose any general matrix $A$ into a positive definite diagonal matrix and distance preserving members of the orthogonal group.

3. Understand the set of alternating multilinear maps

Since the set of alternating $n$-multilinear maps in $n$ dimensions is one dimensional, the determinant is its basis with the property that the determinant of the identity is $1$.

Now we understand multilinear maps as a (weighted) measure of volume change. If we were to measure $k$-volume in $n$-dimensions, we have more possible directions in which we can measure volume, so we stop having only one basis element (the determinant) and instead get multiple. But the principle remains the same. :handwaving: (maybe someone can one up me here)