On affine spaces of alternating matrices with constant rank

Let $\mathbb{F}$ be a field, and $n \geq r>0$ be integers, with $r$ even. Denote by $\mathrm{A}_n(\mathbb{F})$ the space of all $n$-by-$n$ alternating matrices with entries in $\mathbb{F}$. We consider the problem of determining the greatest possible dimension for an affine subspace of $\mathrm{A}_n(\mathbb{F})$ in which every matrix has rank equal to $r$ (or rank at least $r$). Recently Rubei has solved this problem over the field of real numbers. We extend her result to all fields with large enough cardinality. Provided that $n \geq r+3$ and $|\mathbb{F}|\geq \min\bigl(r-1,\frac{r}{2}+2\bigr)$, we also determine the affine subspaces of rank $r$ matrices in $\mathrm{A}_n(\mathbb{F})$ that have the greatest possible dimension, and we point to difficulties for the corresponding problem in the case $n\leq r+2$.