There is a nice formula, called the Plemelj-Smithies formula that allows the computation of the determinant of certain operators in terms of the moments (Tr(A^n)) of A. I'm including this as an answer because the formula organizes the information nicely for all finite dimensions, and holds also in infinite dimensions.

For the formula, see section 1.2 of the paper 

Gohberg, Goldberg and Krupnik *Traces and Determinants of Linear Operators*, Integral Equations and Operator Theory 1996, Volume 26, Issue 2, pp 136-187. 

This paper was eventually extended into a very nice book. 


  [1]: http://krein.unica.it/~cornelis/private/PDF/IEOT/ieot_26_136.pdf