Question

This is mostly a question in combinatorics. Is there a clean way in terms of determinantal identities to write down $c(\wedge^k V)$ i.e. the individual summands in terms of the individual summands of $c(V)$. I am aware of the fact that it involves writing new symmetric polynomials in terms of the elementary ones if one uses splitting principle formalism. But I was just wondering is there are any nice identities in symmetric function theory that allows us to do this without referring to kostka numbers and the matrix $K_{\lambda\mu}$