The load-bearing capacity of slender structural systems, particularly those comprising assemblages of beam elements, is critically governed by their geometric stiffness properties. Conventional methodologies for deriving the geometric stiffness matrix of beam elements are typically rooted in stability functions or variational energy principles and entail mathematically intricate formulations that frequently obscure physical interpretations, especially concerning the treatment of higher-order displacement terms. In marked contrast to these established approaches, the present study introduces a physically insightful framework by first investigating the induced moment matrix. This fundamental mechanical attribute rigorously preserves nodal moment equilibrium during finite three-dimensional rotations. Its formulation is established through the incremental virtual work principle, derived from consistent linearization of three-dimensional solid beam kinematics integrated with exact spatial rotational transformations. A pivotal theoretical finding demonstrates that while the induced moment matrix inherently exhibits asymmetry when examined at the individual element level, this asymmetry vanishes upon assembly into the global structural system, resulting in a symmetric structural-level geometric stiffness matrix. Building upon this foundation, a three-dimensional geometric stiffness matrix incorporating undetermined coefficients is systematically constructed by rigorously enforcing displacement compatibility conditions across the element. Subsequently, by exploiting the proven symmetry of the assembled global geometric stiffness matrix and strictly imposing the rigid body rule, which necessitates zero straining energy for arbitrary rigid displacements, a concise and fully explicit analytical expression for the three-dimensional beam element geometric stiffness matrix is derived. This expression is further simplified to yield its corresponding two-dimensional counterpart for planar analyses. Comprehensive numerical validations encompass eigenvalue buckling analyses of axially compressed prismatic members and geometrically nonlinear analyses of diverse structural configurations: cantilever beams, arches, spatial frames, and curved beams. These extensive simulations consistently confirm that the proposed geometric stiffness matrix achieves exceptional computational accuracy and numerical efficiency in predicting both critical buckling loads and complex post-buckling equilibrium paths. This work establishes a novel, mechanically transparent, and mathematically streamlined paradigm for deriving geometric stiffness matrices, offering a unifying perspective readily extensible beyond beam elements to potentially encompass plate and shell finite elements in future developments.