Affine subspaces of antisymmetric matrices with constant rank

For every $n\in \mathbb{N}$ and every field $K$, let $A(n,K)$ be the vector space of the antisymmetric $(n\times n)$-matrices over $K$. We say that an affine subspace $S$ of $A(n,K)$ has constant rank $r$ if every matrix of $S$ has rank $r$. Define {\cal A}_{antisym}^K(n;r)= \{ S \;| \; S \; \rm affine subspace of $A(n,K)$ of constant rank r\} $$a_{antisym}^K(n;r)=\max \{\dim S\mid S\in {\mathcal{A}}_{antisym}^K(n;r)\}\mathrm{.}$$ In this paper we prove the following formulas: for $n\ge 2r+2$ $$a_{antisym}^{\mathbb{R}}(n;2r)=(n-r-1)r;$$ for $n=2r$ $$a_{antisym}^{\mathbb{R}}(n;2r)=r(r-1);$$ for $n=2r+1$ $$a_{antisym}^{\mathbb{R}}(n;2r)=r(r+1)\mathrm{.}$$