We observe the buildup of strong (approximately 50%) spontaneous vector polarization in emission from a GaN-based polariton laser excited by short optical pulses at room temperature. The Stokes vector of emitted light changes its orientation randomly from one excitation pulse to another, so that the time-integrated polarization remains zero. This behavior is completely different from any previous laser. We interpret this observation in terms of the spontaneous symmetry breaking in a Bose-Einstein condensate of exciton polaritons.