On the dimension of some real, bounded rank, matrix spaces

Given $n$ integer, let $X$ be either the set of hermitian or real $n\times n$ matrices of rank at least $n-1$. If $n$ is even, we give a sharp estimate on the maximal dimension of a real vector subspace of $X\cup\{0\}$. The rusults are obtained, via K-theory, by studying a bundle map induced by the adjugation of matrices