University Police Chief Paul Denton on Tuesday identified the suspect as 51-year-old Brown.
Denton says Brown recently received a poor performance evaluation, but he declined to say if that was a motive for the shooting.
Denton says Brown was pronounced dead at Ohio State University Medical Center several hours after the shooting, which was first reported at 3:30 a.m. Tuesday.
One of the victims, 48-year-old building services manager Larry Wallington, died at the scene. The other is hospitalized in stable condition.
No students were hurt.