Let $V = \mathbb{R}^{2n}$ be the standard representation of the symplectic group $\text{Sp}(2n,\mathbb{R})$, and let $\{a_1,b_1,\ldots,a_n,b_n\}$ be the standard symplectic basis for $V$. Let $$W \subset \left(\wedge^2 V\right)^{\otimes 2}$$ be the $\text{Sp}(2n,\mathbb{R})$-subrepresentation spanned by the elements $$\{\text{$(a_i \wedge b_i) \otimes (a_j \wedge b_j)$ $|$ $1 \leq i,j \leq n$}\}.$$ Is there a nice description of $W$?

More generally, many variants of this question have been arising in my research recently and I'm sure there is some way to answer them with a computer (at least for specific values of $n$). How can I do this?

By the way, I am not a representation theorist, so I apologize if this question is too easy.