General question: We know that the top Chern class $c_n(\xi)$ of an $n$-dimensional complex vector bundle $\xi$ is its Euler class, while the first Chern class, $c_1(\xi)$, is the Euler class of its determinant bundle. Can the intermediate classes also be expressed as Euler classes of bundles related to $\xi$?

Let me also ask a much more specific question: We know that $H^\ast(BU(3);{\mathbb Z}) = {\mathbb Z}[c_1, c_2, c_3]$. Is there a complex 2-plane bundle over the classifying space $BU(3)$ (or $BU(n)$ for $n\geq 3$) whose Euler class is $c_2$?

Motivation: Long story short, I'm trying to construct an analogue of $c_2$ in a context where I know that the usual constructions I'm familiar with won't work. But I do have Euler classes in this context, so if I could express $c_2$ as an Euler class as above, I might be able to use that in my context. On the other hand, if anyone has a proof that there is no such bundle over $BU(3)$, then that will save me time chasing down that particular rabbit hole.