  We give a short proof of the Gauss-Bonnet theorem for a real oriented
Riemannian vector bundle $E$ of even rank over a closed compact orientable
manifold $M$. This theorem reduces to the classical Gauss-Bonnet-Chern theorem
in the special case when $M$ is a Riemannian manifold and $E$ is the tangent
bundle of $M$ endowed with the Levi-Civita connection. The proof is based on an
explicit geometric construction of the Thom class for 2-plane bundles.
