Damping mechanisms in magnetic systems determine the lifetime, diffusion, and transport properties of magnons, domain walls, magnetic vortices, and skyrmions. Based on the phenomenological Landau-Lifshitz-Gilbert equation, here the effective damping parameter in noncollinear magnetic systems is determined describing the linewidth in resonance experiments or the decay parameter in time-resolved measurements. It is shown how the effective damping can be calculated from the elliptic polarization of magnons, arising due to the noncollinear spin arrangement. It is concluded that the effective damping is larger than the Gilbert damping, and it may significantly differ between excitation modes. Numerical results for the effective damping are presented for the localized magnons in isolated skyrmions, with parameters based on the Pd/Fe/Ir(111) model-type system.