rt.representation theory - Can you "Wedge" two representations?

Not an answer, rather an attempt to hijack the question...

Some time ago I have also been wondering how to wedge two vector spaces and came up with the following construction:

Let $f:Uto V$ and $g:Uto W$ be two vector space morphisms. We define the vector space $Vwedge_U W$ (of course, this depends not only on $U$, $V$ and $W$, but also on $f$ and $g$, but we silently leave these out of the notation - just as in the case of fibered products) as the quotient of the tensor product $Votimes W$ by the subspace spanned by all tensors of the form $fleft(uright)otimes gleft(uright)$ for $uin U$.

This is functorial, but does anyone know any use for it? Any results about the structure of $Vwedge_U W$ as a representation, if $U$, $V$ and $W$ are representations? How does this $wedge_U$ operation "look like" in the representation ring (for instance, the usual wedge operations look like the lambda operations $lambda^1$, $lambda^2$, ...).

EDIT: In characteristic $neq 2$, we have $Vwedge_U W=left(Votimes Wright)diagup left(left(left(fotimes gright)circleft(mathrm{id}+tauright)right)left(Uotimes Uright)right)$, where $tau$ is the transposition of the two tensorands. But it is still interesting to find out what exactly is factored out in classical cases, e. g. in representation theory of $S_n$.