Consider matrices with entries in a field $F$ of characteristic $2$. Let $\Omega$ denote the $2n\times2n$ matrix $\left[\begin{array}{ll}0&1_n\\1_n&0\end{array}\right]$. Then $X^t\Omega X$ is symmetric with $0$ diagonal, for each $2n\times2n$-matrix $X$.

Question: can we express each symmetric matrix with zero diagonal in the form $X^t\Omega X$, for some $X$?

Note: This is an simpler version of a question I asked yesterday.