This technical note considers the problem of representing a sufficiently smooth control affine system as a structured potential-driven system and to exploit the obtained representation for stability analysis and state feedback controller design. These problems have been studied in recent years for particular classes of potential-driven systems. To recover the advantages of those representations for the stabilization of general nonlinear systems, the present note proposes a geometric decomposition technique, based on the Hodge decomposition theorem, to re-express a given vector field into a potential-driven form. Using the proposed decomposition technique, stability conditions are developed based on the convexity of a computed potential. Finally, stabilization is studied in the context of the proposed decomposition by reshaping the Hessian matrix of the obtained potential using damping feedback.