Let $k$ be an algebraically closed field with characteristic $\neq 2$, and let $O_n$ be the group of orthogonal matrices, i.e. invertible matrices whose inverse is their transpose. Let $\textrm{GL}_n/O_n$ be the set of left cosets of $O_n$ in $\textrm{GL}_n$. Is there a natural bijection $f$ from $\textrm{GL}_n/O_n$ onto the set of symmetric $n$ by $n$ matrices?
My first guess would be to associate with an $x \in \textrm{GL}_n$ the symmetric matrix $xx^t + x^tx$. Then if $yx^{-1} \in O_n$, then $1 = (x^{-1})^ty^tyx^{-1}$ implies $x^tx = y^ty$, and similarly the fact that $xy^{-1} \in O_n$ implies that $xx^t = yy^t$, so $xx^t + x^tx = yy^t + y^ty$. Thus the mapping $\overline{x} \mapsto xx^t + x^tx$ is at least well defined.
I think I remember hearing somewhere that this mapping can be shown to be surjective. But I am still at a loss to prove that $x,y$ invertible and $xx^t + x^tx = yy^t + y^ty$ implies that $xy^{-1} \in O_n$.
$G/H$ can be identifed with any set $X$ on which 1) $G$ acts transitively 2) such that some $x \in X$ has stabilizer $H$. Similar statements are true if $G, H, X$ are manifolds or varieties. So to identify $GL_n/O_n$ with symmetric matrices we should be trying to find an action of $GL_n$ on symmetric matrices with stabilizer $O_n$.
The relevant action is
$$GL_n \ni G \mapsto \left( X \mapsto GXG^{T} \right)$$
where $X$ is a symmetric matrix. This is the relevant notion of change of coordinates for symmetric matrices interpreted as symmetric bilinear forms. The stabilizer of the identity is all matrices $G \in GL_n$ such that $G G^T = I$, which is precisely $O_n$: this should make sense because the corresponding symmetric bilinear form is the standard "dot product."
From here the remaining question is to show that this action is transitive; this is where you need that $k$ is both algebraically closed and has characteristic not equal to $2$, and here you also need to restrict attention to nondegenerate symmetric matrices or the statement is just false. In terms of symmetric bilinear forms the claim is that with the above hypotheses on $k$, any two nondegenerate such forms are equivalent after a change of coordinates.