  We give an explicit description, in terms of bracket, anchor, and pairing, of
the standard cochain complex associated to a Courant algebroid. In this
formulation, the differential satisfies a formula that is formally identical to
the Cartan formula for the de Rham differential. This perspective allows us to
develop the theory of Courant algebroid connections in a way that mirrors the
classical theory of connections. Using a special class of connections, we
construct secondary characteristic classes associated to any Courant algebroid.
