Polynomial rings and symmetric algebras on the dual module are isomorphic

Question

Question: if $M$ is a finitely generated free R−module of rank n, then $Sym(M^{*}) \cong R[x_{n}, \dots, x_{n}]$, where $M^{*}$ is the dual module of $M$ and $Sym(M^{*}) = \frac{R \oplus M^{*} \oplus (M^{*})^{\otimes 2} \oplus (M^{*})^{\otimes 3} \oplus \dots}{\sim}$. $\sim$ is the submodule generated by $\{m^{1}\otimes m^{2}\otimes\dots\otimes m^{i} - m^{\sigma(1)}\otimes m^{\sigma(2)}\otimes\dots\otimes m^{\sigma(i)}|\sigma \in S_{i}, i \geq 1, m^{j}\in M^{*}\}.$
I am learning to construct vector bundles from modules and saw this conclusion in an article. May I ask if this conclusion is correct? If so, how should it be proven? If not, how can it be modified to be correct.
My attempt: let $\{m^{i}\}^{n}_{i = 1}$ be a set of bases for $M^{*}$ and $\varphi: M^{*} \rightarrow R[x_1, \dots, x_{n}]$, where $\varphi(m^{i})=x_{i}$. This mapping seems to induce isomorphism, but I don't know how to continue proving it.