The length scale of the magnetization gradients in chiral magnets is determined by the relativistic Dzyaloshinskii-Moriya interaction. Thus, even conventional spin-transfer torques are controlled by the relativistic spin-orbit coupling in these systems, and additional relativistic corrections to the current-induced torques and magnetization damping become important for a complete understanding of the current-driven magnetization dynamics. We theoretically study the effects of reactive and dissipative homogeneous spin-orbit torques and anisotropic damping on the current-driven skyrmion dynamics in cubic chiral magnets. Our results demonstrate that spin-orbit torques play a significant role in the current-induced skyrmion velocity. The dissipative spin-orbit torque generates a relativistic Magnus force on the skyrmions, whereas the reactive spin-orbit torque yields a correction to both the drift velocity along the current direction and the transverse velocity associated with the Magnus force. The spin-orbit torque corrections to the velocity scale linearly with the skyrmion size, which is inversely proportional to the spin-orbit coupling. Consequently, the reactive spin-orbit torque correction can be the same order of magnitude as the non-relativistic contribution. More importantly, the dissipative spin-orbit torque can be the dominant force that causes a deflected motion of the skyrmions if the torque exhibits a linear or quadratic relationship with the spin-orbit coupling. In addition, we demonstrate that the skyrmion velocity is determined by anisotropic magnetization damping parameters governed by the skyrmion size. ; Comment: 7 pages