This paper deals with the state estimation problem for stochastic nonlinear differential systems, driven by standard Wiener processes, and presents a filter that is a generalization of the classical Extended Kalman-Bucy filter (EKBF). While the EKBF is designed on the basis of a first order approximation of the system around the current estimate, the proposed filter exploits a Carleman-like approximation of a chosen degree v >= 1. The approximation procedure, applied to both the state and the measurement equations, allows to define an approximate representation of the system by means of a bilinear system, for which a filtering algorithm is available from the literature. Numerical simulations on an example show the improvement, in terms of sample error covariance, of the filter based on the first-order, second-order and third-order system approximations (v = 1, 2, 3).