### how to sum up the function over all permutations of variables in associative non-commutative algebra

hello, i need to sum up $\lambda_{\sigma}(a_{\sigma(1)}a_{\sigma(2)}a_{\sigma(3)}-a_{\sigma(3)}a_{\sigma(4)}a_{\sigma(5)})$ over all $\sigma\in S_5$ where $a_i$ are elements of associative non-commutative algebra.
the result should be $E_1a_1a_2a_3a_4a_5+\cdots+E_{120}a_5a_4a_3a_2a_1$
and i need to express $E_i$ in terms of $\lambda_{\sigma}$.
actualy my tartget is to find non-zero solution of $E_i=0$ for all $i$.