$\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\Sym{Sym}$Let $V = \mathbb{R}^{2n}$ be the standard representation of the symplectic group $\Sp(2n,\mathbb{R})$, and let $\{a_1,b_1,\dotsc,a_n,b_n\}$ be the standard symplectic basis for $V$.  Let
$$W \subset \left(\bigwedge\nolimits^2 V\right)^{\otimes 2}$$
be the $\Sp(2n,\mathbb{R})$-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 $W$?

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EDIT: Here is one small observation that breaks this up into two problems.  The tensor square is the direct sum of the alternating and symmetric tensors, so $W$ is the direct sum of its projections to these two factors.  In other words, what we need to do is determine the $\Sp(2n,\mathbb{R})$-subrepresentation spanned by

$$\{(a_i \wedge b_i) \wedge (a_j \wedge b_j) \mid 1 \leq i < j \leq n\}$$
in $\bigwedge\nolimits^2(\bigwedge\nolimits^2 V))$ and by
$$\{(a_i \wedge b_i) \cdot (a_j \wedge b_j) \mid 1 \leq i,j \leq n\}$$
in $\Sym^2(\bigwedge\nolimits^2 V))$.  We've now reached the limits of my knowledge of representation theory!

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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.