Let $A$ be an $n \times n$ matrix. Are there formulas that convert trace linear combinations of traces of powers of $A$ to determinant of $A$ and vice versa from determinant linear combinations of determinants of powers of $A$ to trace ? of $A$? (I am mainly looking for the latter - conversion of determinants to trace of $A$).

Let $A$ be an $n \times n$ matrix. Are there formulas that convert trace to determinant and vice versa from determinant to trace? (I am mainly looking for conversion of determinants to trace of $A$).