Which even-graded integral cohomology classes arise as Chern classes?

Question

Given a CW complex $X$, is there some way to determine which even-graded elements of its integral cohomology ring $H^\star(X, \mathbb Z)$ arise as Chern classes of a complex vector bundle $E \to X$? I am particularly interested in the case where $X$ is an infinite-dimensional CW complex whose skeleta are all complex manifolds (e.g., direct limits of flag varieties or complex tori) and the restriction of $E$ to each skeleton is a holomorphic vector bundle.