This paper addresses the problem of time-varying bearing formation control in d(d≥2)-dimensional Euclidean space by exploring Persistence of Excitation (PE) of the desired bearing reference. A general concept of Bearing Persistently Exciting (BPE) formation defined in d-dimensional space is here fully developed. By providing a desired formation that is BPE, distributed control laws for multi-agent systems under both single- and double-integrator dynamics are proposed using bearing measurements (along with velocity measurements when the agents are described by double-integrator dynamics), which guarantee uniform exponential stabilization of the desired formation up to a translation. A key contribution of this work is to show that the classical bearing rigidity condition on the graph topology, required for achieving the stabilization of a formation up to a translation and a scale factor, is relaxed and extended in a natural manner by exploring PE conditions imposed either on a specific set of desired bearing vectors or on the whole desired formation. Simulation results are provided to illustrate the performance of the proposed control method.