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](https://mathoverflow.net/questions/180552/multiplicative-symmetrization-of-bilinear-forms) I asked yesterday.



Symmetric Zero-Diagonal Matrices
In fact, the symmetric matrix with zero diagonal over $F$ with $\mathop{\rm char} F=2$ is skew symmetric. It is a standard fact that every skew symmetric (bilinear) form in some basis has matrix $\Omega$ surrounded by zeroes. Each such matrix can be easily obtained from $\Omega$ by an appropriate $X$.


