Variety of maximal isotropic subspaces

Question

Suppose that $V$ is a complex vector space of even dimension $2n$. Let $Q:V \times V \rightarrow \mathbb{C}$ a bilinear, non degenerate, symmetric bilinear form over the field of complex numbers.
Set $$\Sigma=\{ \Lambda \subset C: Q(\Lambda, \Lambda)\equiv 0 \} \subset G(n,2n),$$ the set af all maximal isotropic subspaces of $V$ where $G(n,2n)$ is the grasmannian.
How can I prove that the set $\Sigma$ as a subset of $G(n,2n)$ is a smooth variety and of which dimension?

Answer 1

Hint By construction, the standard action of the special orthogonal Lie group $SO(Q)$ on $V$ induces a transitive action on $\Sigma$.