I'm trying to decompose tensor products of semisimple Lie group/algebra representations into direct sums of irreducible representations. I know this question has been asked many times before, with lots of different answers. See here (Steinberg formula, section 24 in Humphreys, + more options) and here (crystals, Weyl character formula, + more options).
Despite all these resources, I'm not sure where to start and I hope that since my representations aren't too complicated I can use the simplest of these tools.
Say, for example, I wanted to decompose $\wedge^2 V \otimes V^*$ (for $V$ the standard $\mathfrak{sl}_n(\mathbb{C})$ representation). What steps can I take to do this in the easiest way possible?