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Magnetic skyrmions are promising candidates for future spintronic applications such as skyrmion racetrack memories and logic devices. They exhibit exotic and complex dynamics governed by topology and are less influenced by defects, such as edge roughness, than conventionally used domain walls. In particular, their non-zero topological charge leads to a predicted ‘skyrmion Hall effect’, in which current-driven skyrmions acquire a transverse velocity component analogous to charged particles in the conventional Hall effect. Here, we use nanoscale pump–probe imaging to reveal the real-time dynamics of skyrmions driven by current-induced spin–orbit torques. We find that skyrmions move at a well-defined angle ΘSkHthat can exceed 30° with respect to the current flow, but in contrast to conventional theoretical expectations, ΘSkHincreases linearly with velocity up to at least 100 ms−1. We qualitatively explain our observation based on internal mode excitations in combination with a field-like spin–orbit torque, showing that one must go beyond the usual rigid skyrmion description to understand the dynamics.