What's the name (if any) for the multilinear symmetric function that maps a square matrix over some commutative ring $R$ with $1$ to a value in $R$? It's basically like the determinant, but symmetric. It's normalized such that the unit matrix maps to the $1 \in R$.
For example, a $3 \times 3$ matrix would map as follows
$$ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{pmatrix} \mapsto aei + bfg + cdh + ceg + bdi + afh $$