$V \wedge V$ is bad notation when $V$ is a representation, just as $V~ Sym ~V$ would be. $\wedge^2 V$ is less misleading.
You could try to define $V \wedge W = \wedge^2 (V \oplus W)$. This For $v\in V, w\in W$, we can naturally identify $v\wedge w$ and $w \wedge v$ with elements of $\wedge^2 (V \oplus W)$, and $v\wedge w = - w \wedge v$.This has some nice properties, perhaps too trivially, but then be careful that $V\wedge V \ne \wedge^2 V.$
$\wedge^2(V\oplus W) = \wedge^2V \oplus \wedge^2 W \oplus V \otimes W.$
$Sym^2(V\oplus W) = Sym^2 V \oplus Sym^2 W \oplus V \otimes W.$