A computational study is conducted to systematically investigate the effect of rotating speed on vortex shedding characteristics and far-field wake patterns of an elliptic foil rotating in uniform cross flows at low Reynolds numbers. It is found that at low rotating speed, vortices shed at the advancing and retreating edges of the foil could form a strong vortex pair in the wake. At high rotating speed, adjacent vortices are closer to each other, thus may evolve into a single vortex. The far-field wake pattern also strongly depends on the rotating speed and demonstrates complex behaviors. It is observed that the wake can defect from downwards to upwards by increasing the rotating speed. The mean lift force increases monotonically with the rotating speed, while a significant drag reduction occurs at a small rotating speed, which is explained by the corresponding wake pattern. Furthermore, a rotating speed regime exists in which the mean aerodynamic torque is positive so that the rotation of the foil could be solely promoted by the flow, indicating a possibility of flow energy harvesting in this regime.