I want to count the maximum number of rank 2 matrices in a space of certain dimension but I am stuck at some point. Any help/ suggestions are appreciated. Here is the question.

Let $\mathbb{M}_m$ be the space of all skew-symmetric matrices of size $m\geq 4$ over the field $\mathbb{F}_q$ of size $q$ and let $E$ be a subspace of $\mathbb{M}_m$ of dimension $m+2$. Let $E_1$ be a maximal subspace (maximal w.r.t dimension) of $E$ containing only rank 2 matrices and $\dim E_1\leq 4$. Write $E= E_1\bigoplus E_2$ for some subspace $E_2$ of $E$. I want to get an upper bound for the maximum number of rank $2$ matrices in $E$ and I can do that if I could find an upper bound for rank $2$ and $4$ matrices in $E_2$.

Since $\dim E_1$ is small as compared to $m+2$, it seems that $E$ cannot contain many rank $2$ matrices but I am far from a guess.


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