Showing each coset of a group can be associated to a unique symmetric matrix

Question

$f$ is an automorphism where $f: GL(3,ℝ)\rightarrow GL(3,ℝ)$ and $f(A)=(A^{-1})^T$

Let $T(3)$ be the group of fixed points of f. How do we show that for each coset (both left and right) of $T(3)$ there can be associated a unique symmetric matrix?

I see that $T(3)$ basically represents the set of orthogonal matrices $(A^{-1}=A^T)$, but i'm not sure where to go from here.

Answer 1

This follows directly from the polar decomposition as explained in Polar decomposition of real matrices.