Two forms related to the symplectic dual polar space in odd characteristic

Let $V$ be a $2n$-dimensional vector space over a field $\mathbb{F}$ equipped with a non-degenerate alternating form $\xi.$ Let $\mathcal{G}_n$ be the $n$-grassmannian of $\PG(V)$ and $\Delta_n$ the dual of the polar space $\Delta$ associated to $\xi$. Then $\mathcal{G}_n$ and $\Delta_n$ are naturally embedded in the vector space $W_n=\wedge^nV$ and $V_n\subseteq W_n$ respectively, where dim($W_n$)$={ 2n \choose n}$ and dim($V_n$)$={ 2n \choose n}-{ 2n \choose {n-2}}.$ The spaces $W_n$ and $V_n$ can be regarded as modules for the symplectic group $Sp(2n, \mathbb{F}).$ If char($\mathbb{F}$)$\not= 2$, we will define two forms $\alpha$ and $\beta$ of $W_n$ which coincide on $V_n$ and we will investigate the relation between these two forms and the collineation of $W_n$ naturally induced by $\xi$. We will obtain a description of the module $W_n$ in terms of the two subspaces of $W_n$ where the linear functionals induced by $\alpha$ and $\beta$ are equal and respectively opposite.\\