Abstract
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
2
n−1
times the Euler class under the usual class map from the Chow ring to ordinary cohomology. Moreover, we show this class represents
1/
2
n−1
(n−1)!
times the
n
th
Chern class of the representation of SO(2n) whose highest weight vector is twice that of the half-spin representation.