I need to decompose a specific $SO(n)$-representation into irreducible ones, but my background on representation theory is rather poor, so I would like to post the problem here.
Let $V=\mathbb{R}^n$ with the natural action of $SO(n)$. This induces an action $\rho$ of $SO(n)$ on $\bigwedge^r V$. Now we consider the action on $\textrm{Hom}(V,\bigwedge^r V)$ by defining $(g.\varphi)(v):=\rho(g)(\varphi(g^{-1}v))$ for all $g\in SO(n)$ and $v\in V$.
Question: How does $\textrm{Hom}(V,\bigwedge^r V)$ decompose into irreducible $SO(n)$-submodules?
Here is how far I got: For $r=0$ this is already irreducible and for $r=1$ we can identify our space with the space of $n\times n$ matrices. This decomposes into the space spanned by the identity, the traceless symmetric matrices and the skew-symmetric matrices. For the latter I am not 100% sure if this is in fact an irreducible representation and for $r>1$ I don't have a clue.