Let $E=\xi-\eta$ be a virtual vector bundle over a compact base $B$, which we may assume is a CW complex. A quick and dirty way to define the total Stiefel-Whitney class $w(E)\in H^\ast(B;\mathbb{Z}/2\mathbb{Z})$ would be to say that, morally, $E\oplus\eta\cong \xi$, and so we can use the Whitney product formula $w(E\oplus\eta)=w(E)w(\eta)$ to get $$w(E) = w(\xi)w(\eta)^{-1},$$ the element $w(\eta)\in H^\ast(B;\mathbb{Z}/2\mathbb{Z})$ being invertible since $B$ is compact.

I am interested in a more righteous definition, along the lines of Whitney's original definition for honest bundles in terms of obstructions to finding linearly independent sections over skeleta. I think that such a definition exists (maybe in terms of sectioning a certain Hom-bundle) but I wasn't able to find a reference. Hence I ask:

Can the Stiefel-Whitney classes of a virtual vector bundle be defined obstruction theoretically, and if so, is there a good reference in the literature describing this construction?