This technical note deals with the global asymptotic stabilization problem for a class of bilinear systems. A state feedback controller solving this problem is obtained uniting a local controller, having an interesting behavior in a neighborhood of the origin, and a constant controller valid outside this neighborhood. The approach developed is based on the use of a hybrid loop, and more precisely a hybrid state feedback. This result is extended to the case where the state of the plant is not fully available and only the measured output can be used for control purposes. In this case, a dynamical controller constituted by an observer and a state feedback is built by means of hybrid output feedback framework. In both cases, the conditions are expressed by a set of linear matrix inequalities.