Let ${{M}_{n}}(F)$ be the algebra of $n\times n$ matrices over a field $F$ of characteristic $p>2$ and let $*$ be an involution on ${{M}_{n}}(F)$. If ${{s}_{1}},...,{{s}_{r}}$ are symmetric variables we determine the smallest $r$ such that the polynomial

$${{P}_{r}}({{S}_{1}},...,{{S}_{r}})\,=\,\sum\limits_{\sigma \in {{S}_{r}}}{{{S}_{\sigma (1)}}...{{S}_{\sigma (r)}}}$$

is a $*$-polynomial identity of ${{M}_{n}}(F)$ under either the symplectic or the transpose involution. We also prove an analogous result for the polynomial

$${{C}_{r}}\left( {{k}_{1}},...,{{k}_{r,}}{{{{k}'}}_{1}},...,{{{{k}'}}_{r}} \right)=\sum\limits_{\sigma ,\tau \in {{S}_{r}}}{{{k}_{\sigma \left( 1 \right)}}{{{{k}'}}_{\tau \left( 1 \right)}}\cdot \cdot \cdot {{k}_{\sigma \left( r \right)}}{{{{k}'}}_{\tau \left( r \right)}}}$$

where ${{k}_{1}},...,{{k}_{r}},k_{1}^{'},...,k_{r}^{'}$ are skew variables under the transpose involution.