Large 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. In a recent work, we have determined the maximal dimension for a linear subspace of $n$ by $n$ symmetric matrices with rank less than or equal to $r$, and we have classified the spaces having that maximal dimension. In this article, provided that $\mathbb{K}$ has more than two elements, we extend this classification to spaces whose dimension is close to the maximal one: this generalizes a result of Loewy. We also prove a similar result on spaces of alternating matrices with bounded rank, with no restriction on the cardinality of the underlying field.