  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.
