Generation, detection and manipulation of pure spin currents in semiconductors play a major role in the realisation of the next generation of spin-based (nano)electronic elements. The spin-orbit interaction (SOI) in III-nitride semiconductors, as well as the long spin relaxation times expected in these materials, indicate them as appealing hosts for spin currents and related phenomena. Spin pumping in non-magnetic materials and particularly in semiconductors with Rashba SOI allows to establish spin currents and to convert them into charge current via the inverse spin Hall effect. In this work, the design, fabrication, and characterisation of a permalloy (Py)/n-GaN:Si bilayer system are reported and pure spin current is generated through spin pumping from Py into n-GaN:Si at room temperature. The onset of spin pumping is detected in conditions of ferromagnetic resonance, together with the produced Hall voltage. Furthermore, in a series of samples with thickness of the n-GaN:Si between 150 and 1900 nm, the effect of the n-GaN:Si layer thickness on the efficiency of the mechanism is investigated. For n-GaN:Si thinner than 1 µm, spin backflow current dominates and no generated voltage may be measured. The spin-Hall angle ?(SH) provides a measure of the spin-to-charge-current conversion efficiency. In the system under consideration - and upon eliminating the spurious components which contribute to the generated voltage - we have found for ?(SH) a value of the order of (3.03x10^{-3}), i.e. at least one order of magnitude higher than those reported for other technologically significant semiconductors, opening wide perspectives for III-nitrides as efficient spin current generator materials.