Additive Maps of Rank k Bivectors

Let ${\cal U}$ and ${\cal V}$ be linear spaces over fields $\mathbb{F}$ and $\mathbb{K}$, respectively, such that Dim$\,{\cal U}=n\geqslant 2$ and $\left|\mathbb{F}\right|\geqslant 3$. Let $\bigwedge^2{\cal U}$ be the second exterior power of ${\cal U}$. Fixing an even integer $k$ satisfying $\frac{n-1}{2}\leqslant k\leqslant n$, it is shown that a map $\psi:\bigwedge^2{\cal U}\rightarrow\bigwedge^2{\cal V}$ satisfies $\psi(u+v)=\psi(u)+\psi(v)$ for all rank $k$ bivectors $u,v\in\bigwedge^2{\cal U}$ if and only if $\psi$ is an additive map. Examples showing the indispensability of the assumption on $k$ are given.