We consider nonlinear dynamical systems driven by stochastic forcing. It has been largely evidenced in the literature that the linear response of non-normal systems (e.g. fluid flows) may exhibit a large variance amplification, even in a linearly stable regime. This linear response, however, is relevant only in the limit of vanishing forcing intensity. As the intensity increases, the frequency distribution and the variance of the response may be strongly modified due to nonlinear effects. To go beyond this limitation, we propose a theoretical approach to derive an amplitude equation governing the weakly nonlinear evolution of these systems. This approach does not rely on the presence of an eigenvalue close to the neutral axis, applying instead to any sufficiently non-normal operator, and the Fourier components of the response are allowed to be arbitrarily different from any eigenmode. The methodology is outlined for a generic nonlinear dynamical system, and the application case highlights a common non-normal mechanism in hydrodynamics: convective non-normal amplification in the flow past a backward-facing step.