Problem concerning about an $n$-subspace of $ A_{n}(F) $

Question

Let $A_{n}(F) $ denote the $n \times n$ skew symmetric matrices over a finite field $F$. Suppose $n$ be even and $N$ be a subspace of $A_{n}(F) $. Now if all the non-zero matrices in $N$ are invertible, then the maximum the dimension of $N$ will be $n/2$. The upper bound of this maximum dimension follows from Chevalley warning theorem . Also I know that there is a such type of subspace of dim $n/2$. But the way to get this result is tricky. Does there any simplest process to find this type subspace $N$ of $\dim n/2$? Provide me some example of such type subspace.

Answer 1

If you take $n/2 \times n/2$ matrix $A$, with an irreducible minimal polynomial (as can be constructed with a companion matrix), the subspace $W = \text{span}\left(I, A, \ldots, A^{n/2-1}\right)$ will be a subspace of matrices of dimension $n/2$ where all non-zero matrices are invertible.

You can create the subspace of $n \times n$ skew symmetric matrices by taking matrices of the form

$$\left[\begin{array}{cc}0 & X \\ -X & 0 \end{array}\right]$$

for $X \in W$.