This is probably a very well-known representation theory / module theory question, so please excuse my ignorance in certain branches of mathematics. But consider the action of $SO(n)$ on $M_{n \times n}(\mathbb{R})$, the set of $n \times n$ matrices. How does this action decompose into irreducible ones? I am particularly interested in the orbit of the identity $I_n \in M_{n \times n}$ under the action of $\mathbb{R} SO(n)$.