We investigate the current-induced switching of the Néel order in NiO(001)/Pt heterostructures, which is manifested electrically via the spin Hall magnetoresistance. Significant reversible changes in the longitudinal and transverse resistances are found at room temperature for a current threshold lying in the range of 10^{7} A/cm^{2}. The order-parameter switching is ascribed to the antiferromagnetic dynamics triggered by the (current-induced) antidamping torque, which orients the Néel order towards the direction of the writing current. This is in stark contrast to the case of antiferromagnets such as Mn_{2}Au and CuMnAs, where fieldlike torques induced by the Edelstein effect drive the Néel switching, therefore resulting in an orthogonal alignment between the Néel order and the writing current. Our findings can be readily generalized to other biaxial antiferromagnets, providing broad opportunities for all-electrical writing and readout in antiferromagnetic spintronics.