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We present low-temperature magnetoconductivity measurements of a density-tunable and high mobility two-dimensional electron gas confined in the wide band-gap $\mathrm{Ga}\mathrm{N}∕\mathrm{Al}\mathrm{Ga}\mathrm{N}$ system. We observed pronounced antilocalization minima in the low-field conductivity, indicating the presence of strong spin-orbit coupling. Density dependent measurements of magnetoconductivity indicate that the coupling is mainly due to the Bychkov-Rashba mechanism. In addition, we have derived a closed-form expression for the magnetoconductivity, allowing us to extract reliable transport parameters for our devices. The Rashba spin-orbit coupling constant is ${\ensuremath{\alpha}}_{\mathrm{so}}\ensuremath{\sim}6\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}\phantom{\rule{0.3em}{0ex}}\mathrm{eV}\phantom{\rule{0.2em}{0ex}}\mathrm{m}$, while the conduction-band spin-orbit splitting energy amounts to ${\ensuremath{\Delta}}_{\mathrm{so}}\ensuremath{\sim}0.3\phantom{\rule{0.3em}{0ex}}\mathrm{meV}$ at ${n}_{e}=1\ifmmode\times\else\texttimes\fi{}{10}^{16}\phantom{\rule{0.3em}{0ex}}{\mathrm{m}}^{\ensuremath{-}2}$.