Orbit closures of symmetric bilinear form

Question

Let $A$ and $B$ be two real symmetric matrices in $M_n(\mathbb{R})$. I would like to learn about necessary and sufficient conditions for knowing when $B \in \overline{GL_n(\mathbb{R})\cdot A}$; where: $$ GL_n(\mathbb{R})\cdot A:=\{(g^{-1})^{T} A g^{-1} : g \in GL_n(\mathbb{R})\}, $$ $(g^{-1})^{T}$ is the transpose of $g^{-1}$ and $\overline{GL_n(\mathbb{R})\cdot A}$ is the closure of $GL_n(\mathbb{R})\cdot A$ with respect to the usual topology of $M_n(\mathbb{R})$.

Help would be appreciated!

Thanks!