This paper addresses the problem of bearing formation control in $d$ $(d\ge 2)$-dimensional 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 fully developed for the first time. 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 (also velocity measurements for double-integrator dynamics), which guarantee exponential stabilization of the desired formation up to a translation vector. 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 scaling factor, is relaxed in a natural manner by exploring PE conditions imposed solely on a specific set of desired bearing vectors. Simulation results are provided to illustrate the performance of the proposed control method.