lie algebras - Invariant forms

Not a full answer, but you can get plenty of examples by taking $V$ to be the coadjoint representation of a Lie algebra with nonzero Killing form. Then $omega in Lambda^3mathfrak{g}^*$, defined by
$$omega(X,Y,Z) = mathrm{Tr}~mathrm{ad}([X,Y])mathrm{ad}(Z),$$
is invariant and nonzero.

Similarly, any metric Lie algebra with $V$ the adjoint representation also works. The invariant 3-form is then
$$omega(X,Y,Z) = langle [X,Y], Zrangle,$$
with $langle-,-rangle$ the ad-invariant inner product.