We report the first full-dimensional potential energy surface (PES) and quantum mechanical close-coupling calculations for scattering of SiO due to H2. The full-dimensional interaction potential surface was computed using the explicitly correlated coupled-cluster (CCSD(T)-F12b) method and fitted using an invariant polynomial approach. Pure rotational quenching cross sections from initial states v1 = 0, j1 = 1-5 of SiO in collision with H2 are calculated for collision energies between 1.0 and 5000 cm-1. State-to-state rotational rate coefficients are calculated at temperatures between 5 and 1000 K. The rotational rate coefficients of SiO with para-H2 (p-H2) are compared with previous approximate results which were obtained using SiO-He PESs or scaled from SiO-He rate coefficients. Rovibrational state-to-state and total quenching cross sections and rate coefficients for initially excited SiO (v1 = 1, j1 = 0 and 1) in collisions with p-H2 (v2 = 0, j2 = 0) and ortho-H2 (o-H2) (v2 = 0, j2 = 1) are also obtained. The application of the current collisional rate coefficients to astrophysics is briefly discussed.