How to build the principal SU(2) bundles on surfaces?

Is there a way to classify (and build) the principal SU(2) bundles over a given topological surface up to homeomorphism? In the end, I would like to examine the associated bundle whose fiber is a given irreducible representation of SU(2). Perhaps this construction appears in the physics literature.

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Principal $G$-bundles are classified up to isomorphisms by homotopy classes of mappings to $BG$ or by elements of a non-abelian cohomology set. I've tried to summarize this here: mathoverflow.net/questions/15224/triviality-of-fibre-bundles/… –  algori Feb 22 '10 at 1:47

By "surface" you mean a 2-real-dimensional manifold $F$? If so, there is only one principal $SU(2)$ bundle up to bundle isomorphism, the trivial bundle: $F\times SU(2)\to F$. This is because the 3 skeleton of $BSU(2)$ is a point. So the associated bundles to any rep $SU(2)\to GL(V)$ are also trivial: $F\times V$.