Let $V$ be a real vector space with dimension $2n$ with a canonical inner product $\langle\cdot,\cdot\rangle$.
Let $\Omega\in\bigwedge^2V^*$ be a non-degenerated $2$-form on $V$.
I'm trying to prove the following:
There is an orthonormal basis $\{e_1,...,e_{2n}\}$ of $V$ for which the matrix representation of $\Omega$ is block-diagonal with blocks $\left(\begin{matrix}0 & \lambda_i\\ -\lambda_i & 0\end{matrix}\right)$, $i=1,...,n$.
My initial idea was to think of $\Omega$ as a sympletic form and use the classical theorem for the existence of a sympletic basis, i.e., a basis $\{e_1,f_1,...,e_n,f_n\}$ for which $\Omega$ is represented by a block diagonal with blocks $\left(\begin{matrix}0 & 1\\ -1 & 0\end{matrix}\right)$.
Furthermore, normalizing each $e_i's$ and $f_i's$ if necessary, we may assume the basis $\{e_1,f_1,...,e_n,f_n\}$ to be composed of unit vectors and $\Omega$ to be represented by blocks $\left(\begin{matrix}0 & \lambda_i\\ -\lambda_i & 0\end{matrix}\right)$ with $\lambda_i\neq 0$.
The thing is: $\{e_1,f_1,...,e_n,f_n\}$ may not be orthonormal, and I don't know how to fix that.
Any suggestions?