For n a power of two, we give a complete description of the cohomology
algebra H∗(Gn,3;Z2) of the Grassmann manifold
Gn,3 of oriented 3-planes in Rn. We do this by
finding a reduced Gr\"obner basis for an ideal closely related to this
cohomology algebra. Using this Gr\"obner basis we also present an additive
basis for H∗(Gn,3;Z2)