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 [here][1]. This is a paper by Gohberg, Goldberg and  Krupnik that eventually they extended into a very nice book. 

Later on, I'll try to come back and type in the formula... 


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