We calculate exactly in closed analytic form the Coulomb electrostatic potential created by a uniformly charged square plate at an arbitrary point in space. We apply a suitable transformation of variables that allows us to obtain a simple presentation of the Coulomb electrostatic potential in one-dimensional integral form. The final closed form analytic expression of the electrostatic potential is obtained after completing the calculation of the resulting one-dimensional integrals. The exact analytic formula that we derive reflects the symmetry of the square and is easy to implement. The exact analytical result reported in this work represents a much desirable compact, symmetric and easy to implement expression for the Coulomb electrostatic potential created by a uniformly charged square plate at an arbitrary point in space.