  In this paper, we consider the exterior algebra $\Lambda(W)$ of a polynomial
$\mathrm{GL}(n)$-module $W$ and use previously developed methods to determine
the Hilbert series of the algebra of invariants $\Lambda(W)^G$, where $G$ is
one of the classical complex subgroups of $\mathrm{GL}(n)$, namely
$\mathrm{SL}(n)$, $\mathrm{O}(n)$, $\mathrm{SO}(n)$, or $\mathrm{Sp}(2d)$ (for
$n=2d$). Since $\Lambda(W)^G$ is finite dimensional, we apply the described
method to compute a lot of explicit examples. For
$\Lambda(S^3\mathbb{C}^3)^{\mathrm{SL}(3)}$, using the computed Hilbert series,
we obtain an explicit set of generators.
