The dipole anisotropy seen in the {cosmic microwave background radiation} is interpreted as due to our peculiar motion. The Cosmological Principle implies that this cosmic dipole signal should also be present, with the same direction, in the large-scale distribution of matter. Measurement of the cosmic matter dipole constitutes a key test of the standard cosmological model. Current measurements of this dipole are barely above the expected noise and unable to provide a robust test. Upcoming radio continuum surveys with the SKA should be able to detect the dipole at high signal to noise. We simulate number count maps for SKA survey specifications in Phases 1 and 2, including all relevant effects. Nonlinear effects from local large-scale structure contaminate the {cosmic (kinematic)} dipole signal, and we find that removal of radio sources at low redshift ($zlesssim 0.5$) leads to significantly improved constraints. We forecast that the SKA could determine the kinematic dipole direction in Galactic coordinates with an error of $(Delta l,Delta b)sim(9^circ,5^circ)$ to $(8^circ, 4^circ)$, depending on the sensitivity. The predicted errors on the relative speed are $sim 10%$. These measurements would significantly reduce the present uncertainty on the direction of the radio dipole, and thus enable the first critical test of consistency between the matter and CMB dipoles.
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