Symmetrization map

Question

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra, define a symmetrization map $S\mathfrak{g} \to U\mathfrak{g}$ by $x_1 .. x_n \to \frac{1}{n!} \sum x_{\sigma(1)} .. x_{\sigma(n)}$ (here $x_j$ are elements of degree 1). As far as I know, it is an isomorphism. How to write down the inverse map?