In the present study, phase-locked tomographic particle image velocimetry measurements are performed to obtain the complex three-dimensional vortex system created by the interaction of plasma synthetic jets with external crossflow. Three orifice configurations (round, transverse slot, and longitudinal slot) are investigated. For the round orifice case, the vortex system consists of a starting vortex ring surrounding the jet head, a hanging vortex pair residing in the two lateral sides of the jet body, several shear layer vortices bridging the two legs of the hanging vortex pair, and a hairpin vortex induced by the low-speed secondary jet. For the slot orifice cases, the above vortex system is also present; nevertheless, the interconnections of the vortices are further intersected by the rib vortices that are branched out of the elongated vortex ring during axis switching. The counter-rotating vortex pair observed in the far field is essentially evolving from the hanging vortex pair in the near field.