The Chow ring of the classifying space BSO(2n,C)

We compute the Chow ring of the classifying space BSO(2n,C) in the sense of Totaro using the fibration Gl(2n)/SO(2n)→BSO(2n)→BGl(2n) and a computation of the Chow ring of Gl(2n)/SO(2n) in a previous paper. We find this Chow ring is generated by Chern classes and a characteristic class defined by Edidin and Graham which maps to 2n−1 times the Euler class under the usual class map from the Chow ring to ordinary cohomology. Moreover, we show this class represents 1/2n−1(n−1)! times the nth Chern class of the representation of SO(2n) whose highest weight vector is twice that of the half-spin representation.