Smoothness is an etale local property, and you can check that symmetric power sends etale maps to etale maps. So checking on $\mathbb{C}^n$ is fine, since all smooth varieties are etale locally isomorphic to it.

Of course, this is secretly the same as VA's answer, but I like the sound of it better.

Smoothness is an etale local property, and you can check that symmetric power sends etale maps to etale maps. So checking on $\mathbb{A}^n$ is fine, since all smooth varieties over an algebraically closed field is etale locally isomorphic to it.

Of course, this is secretly the same as VA's answer, since stuff which makes sense etale locally is exactly the stuff remembered by the completion of the local ring, but I like the sound of it better.