Skyrmions are magnetic bubbles with nontrivial topology envisioned as data bits for ultrafast and power efficient spintronic memory and logic devices. They may be stabilized in heavy metal/ferromagnetic/oxide trilayer systems. The skyrmion chirality is then determined by the sign of interfacial Dzyaloshinskii-Moriya interaction (DMI). Nevertheless, for apparently identical systems, there is some controversy about the DMI sign. Here we show that the degree of oxidation of the top interface and the thickness of the ferromagnetic layer play a major role. Using Brillouin light scattering measurements in Ta/FeCoB/TaO$_x$ trilayers, we have demonstrated a sign change of the DMI with the degree of oxidation of FeCoB/TaO$_x$ interface. Using polar magneto-optical Kerr effect microscopy, we consistently observed a reversal of the direction of current-induced motion of skyrmions with the oxidation level of TaOx, attributed to their chirality reversal. In addition, a second chirality reversal is observed when changing the FeCoB thickness, probably due to the proximity of the two FeCoB interfaces in the ultrathin case. By properly tuning the chirality of the skyrmion, the spin transfer and spin orbit torques combine constructively to enhance the skyrmion velocity. These observations thus allow envisioning an optimization of the material parameters producing highly mobile skyrmions in this system and could be extended to other stacks. This chirality control also enables a more versatile manipulation of skyrmions and paves the way towards multidirectional devices.