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We present a beam summation (BS) representation for the field scattered by an impedance wedge illuminated by a general 3D electromagnetic Gaussian beam (EM-GB). The emphasis here is not only on the solution of the beam diffraction problem, but mainly on the BS representation. In this representation, the field is expressed as a beam optics (BO) term plus an edge field, described as a sum of diffracted EM-GB's emerging from a discrete set of points and directions along the edge. We introduce an edge-fixed set of EM-GB's that provides a basis for the edge field. The expansion coefficients (the beam's excitation amplitudes) account in a dyadic format for the polarization of the incident beam and also for its direction, displacement from the edge, collimation, and astigmatism. We derive exact expressions for these coefficients as well as simpler approximations that are valid uniformly as a function of the incident beam distance from the edge. The results of this paper provide essential building blocks for a BS representation of EM fields in complex configurations, where the source excited field is described as a sum of beam propagators, and the diffracted fields generated by propagators that hit near edges are also described using beams.