Magnetic skyrmions are topological protected nanometric swirling spin structures, and usually possess a fixed chiral feature. Herein, we theoretically demonstrate a dynamic switching for the chirality in magnetic skyrmions by optimizing the spin-transfer torque (STT). Such a switching process is coupled with the breathing mode, presented as an auto-oscillation phenomenon. In this case, the STT plays a role as the compensation of the intrinsic magnetic damping over a period of oscillation. The oscillation frequency could be tuned by the applied current density and type, and the material parameters such as the anisotropy constant and ferromagnetic exchange coefficient. Our results would be significant for designing nanoscale spin-transfer nano-oscillators.