If by 'like the above form' you mean an expression in terms of radicals of the traces of the powers of $M$, the answer is 'no' for $n>3$. Even for $n=3$, where there is such an expression, it is very ugly and will almost certainly be useless for any practical purpose:
In the case $n=3$, let $p_i = \mathrm{tr}(M^i)$ for $i=1,2,3$, and let $q = \bigl(\|A\|_*\bigr)^2$. Then the relation between $p_1$, $p_2$, $p_3$, and $q$ is found to be $$ 0 = q^4 - 4p_1\,q^3 + (4p_2{+}2p_1^2)\,q^2 + (24p_1p_2{-}{\tfrac{20}{3}}p_1^3{-}{\tfrac{64}{3}}p_3)\,q + (p_1^4{-}4p_1^2p_2{+}4p_2^2), $$ so one can solve for $q$ in radicals using the quartic equation and then take the square root to get $\|A\|_*$. That's the 'simplest' explicit analytical expression.
For higher $n$, there will still be such a polynomial, but of higher degree in $q$, and there won't be a solution in radicals.