Affine spaces of symmetric or alternating matrices with bounded rank

Let $r$ and $n$ be positive integers such that $r<n$, and $\mathbb{K}$ be an arbitrary field. We determine the maximal dimension for an affine subspace of $n$ by $n$ symmetric (or alternating) matrices with entries in $\mathbb{K}$ and with rank less than or equal to $r$. We also classify, up to congruence, the subspaces of maximal dimension among them. This generalizes earlier results of Meshulam, Loewy and Radwan that were previously known only for linear subspaces over fields with large cardinality and characteristic different from $2$.