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Suppose you have a trivial bundle of rank $k$ inside a trivial bundle of rank $n$. Then you get an obvious map to the Grassmannian of $k$-planes in $\mathbb{R}^n$. This map is homotopic to a constant map just when the quotient bundle is trivializable. So the homotopy type of that map is your invariant. You can say something about that homotopy type (at least stably) in the form of characteristic classes, i.e. by pulling back the cohomology of the Grassmannian by the map.