I am looking, without luck so far, for references for an equivariant Gram-Schmidt.
Specifically, Let $P$ be the permutation matrices, and $D$ the diagonal matrices with entries in $\mathbb R^\times$. Then $P,D$ have the obvious right actions on $GL(n,\mathbb R)$, and the coset spaces are (respectively) the space of unordered bases and what we might call the projective frames, i.e., $n$ maximally non-degenerate lines. We can take the coset space of the closure of these two groups acting on the right, too, to get unordered projective bases.
Each of the above has an orthogonal version as a subset, which is the quotient of $O(n)$ by the given right action.
One can argue, fairly straightforwardly, that inclusion of the orthogonal version in the general version is a homotopy equivalence cofibration and hence a deformation retraction exists in the other direction. One should also be able to prove this by arguing that $\sum_{i\ne j}|v_i\cdot v_j|/|v_i||v_j|$ is an equivariant Morse-Bott function, with critical points exactly the orthogonal version.
Anyway, this deformation retraction (perhaps even explicitly given) seems like something which should be well-known. Might anyone have a reference?