  We discuss secondary (and higher) characteristic classes for algebraic vector
bundles with trivial top Chern class. We show that if X is a smooth affine
scheme of dimension d over a field k of finite 2-cohomological dimension (with
char(k) $\neq$ 2) and E is a rank d vector bundle over X, vanishing of the
Chow-Witt theoretic Euler class of E is equivalent to vanishing of its top
Chern class and these higher classes. We then derive some consequences of our
main theorem when k is of small 2-cohomological dimension.
