The structure of symmetric powers of finite-dimensional local rings

Question

Fix an algebraically closed field $k$ of arbitrary characteristic $p$ and let $R$ be a finite-dimensional local $k$-algebra (so in particular $R$ is Artinian and Noetherian). Let $S_n$ be the symmetric group on $n$ elements. Then we have the $n^{th}$ symmetric power $R^{(n)}$ of $R$ which is defined by $$ R^{(n)} := (R^{\otimes n})^{S_n}. $$ Here, by $(-)^{S_n}$ I mean the $S_n$-invariants ($S_n$ has a natural action on $R^{\otimes n}$ by $k$-algebra automorphisms). I'm interested in understanding the structure of these rings - for example, given a particular $R$, is it easy to explicitly describe $R^{(n)}$? I'm happy to assume that $p > n$ if necessary. I assume that this is well-known to experts, but I don't know where to look for references. [Remark that it is not difficult to find information on symmetric powers of $k$-varieties, but in this case spec $R$ is not in general a $k$-variety.]