We simulate the switching behavior of nanoscale synthetic antiferromagnets (SAFs), inspired by recent experimental progress in spin-orbit-torque switching of crystal antiferromagnets. The SAF consists of two ferromagnetic thin films with in-plane biaxial anisotropy and interlayer exchange coupling. Staggered field-like Rashba spin-orbit torques from the opposite surfaces of the SAF induce a canted net magnetization, which triggers an orthogonal torque that drives 90$^circ$ switching of the Neel vector. Such dynamics driven by the field-like spin-orbit torque allows for faster switching with increased Gilbert damping, without a significant detrimental increase of the threshold switching current density. Our results point to the potential of SAFs as model systems, based on simple ferromagnetic metals, to mimic antiferromagnetic device physics.