Wurtzite GaN grown onto an $A$-plane sapphire exhibits a uniaxial strain because the thermal expansion coefficient of the substrate is anisotropic. Measuring the dependence of the transition energies and of the oscillator strengths on the polarization of light, we deduced the value of the spin-exchange energy in wurtzite GaN: $\gamma \approx 0.6\pm 0.1$ meV.