Maybe if I unravel this enough it is actually circular, but I don't think so.

Given an element of $M \in \textrm{GL}_{n+1}(\mathbb{R})$, restrict it to the sphere $\tilde{M}: S^n \to S^n$ via $\tilde{M}(p) = \frac{M(p)}{|M(p)|}$.

We can prove that $\pi_n(S^n) \cong \mathbb{Z}$ directly using Hurewicz and computing $H_n(S^n)$. None of this (I think) requires the sign of a permutation to be defined.

We now just need to check that $\tilde{I}$ gets mapped to $1 \in \mathbb{Z}$ while the permutation matrix of a transposition gets sent to $-1$.

EDIT: This doesn't work, since orientation is built into the definition of homology.

Maybe if I unravel this enough it is actually circular, but I don't think so.

Given an element of $M \in \textrm{GL}_{n+1}(\mathbb{R})$, restrict it to the sphere $\tilde{M}: S^n \to S^n$ via $\tilde{M}(p) = \frac{M(p)}{|M(p)|}$.

$H_n(S^n) \cong \mathbb{Z}$ by an easy computation. Now just need to check that $\tilde{I}$ gets mapped to $1 \in \mathbb{Z}$ while the permutation matrix of a transposition gets sent to $-1$.