I guess that the image (all matricies of rank $2r$ with $r$ positive and $r$ negative eigenvalues) of the map
$$\psi: GL_n(\mathbb{C}) \to M_{n,n}(\mathbb{C}) \qquad A \mapsto A(-1_r \oplus {1}_r \oplus 0_{n-2r})A^*$$
is a manifold of dimension $4nr-4r^2$.
Maybe if I can show that $\psi$ is an embedding the problem is solved.
It should be clear that $\psi$ is injective but I don't know how to prove that it is open.