I would like an explicit description of $\mathbb{R} SO(n) I_n$, i.e., the image of the identity under the action of the group algebra of $SO(n)$. Presumably this is well-known and in the context of all compact simple Lie groups?

I would like an explicit description of $\mathbb{R} SO(n) I_n$, i.e., the image of the identity under the action of the group algebra of $SO(n)$ by left multiplication. Equivalently, what is an explicit description of the vector space $\{\sum_i c_i A_i: A_i \in SO(n)\}$? Presumably this is well-known and in the context of all compact simple Lie groups? I need this to understand some continuous state space Markov chain of particle systems.

Edit: to those who vote to close, please state reason. If you have a one-liner answer, why not give it a shot? I will close it myself when I see a satisfactory response.

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 $\mathbb{R} SO(n)$.

I would like an explicit description of $\mathbb{R} SO(n) I_n$, i.e., the image of the identity under the action of the group algebra of $SO(n)$. Presumably this is well-known and in the context of all compact simple Lie groups?