  Microcavity polaritons in the lasing regime undergo a spontaneous symmetry
breaking transition resulting in coherent emission with a well defined
polarization. The order parameter is thus a vector describing both the laser
global phase and polarization. Using an ultrafast single-shot detection
technique we show that polariton lasing in GaAs-based microcavities presents a
high degree of second order coherence ($g^{(2)}(\tau=0) \approx 1$) above
threshold, and that the initial polarization is stochastic, taking any possible
direction in the Poincar\'e sphere (linear, elliptical or circular). Once the
polarization direction is established, subsequent oscillations of the emission
probability witness the presence of an intrinsic polarization splitting. Our
results show the negligible role of polariton interactions in the total
emission statistics and in the establishment of the initial polarization.
