  Let GO(2n) be the general orthogonal group (the group of similitudes) over
any algebraically closed field of characteristic not equal to 2. We determine
the etale cohomology ring with mod 2 coefficients of the algebraic stack
BGO(2n). In the topological category, Y. Holla and N. Nitsure determined the
singular cohomology ring of the classifying space BGO(2n) of the complex Lie
group GO(2n) in terms of explicit generators and relations. We extend their
results to the algebraic category. The chief ingredients in this are (i) an
extension to etale cohomology of an idea of Totaro, originally used in the
context of Chow groups, which allows us to approximate the classifying stack by
quasi projective schemes; and (ii) construction of a Gysin sequence for the G_m
fibration BO(2n) to BGO(2n) of algebraic stacks.
