The general theory of relativity fundamentally relies on the metric tensor to describe gravitational phenomena, traditionally eschewing the vector field formulations ubiquitous in electrodynamics and quantum field theories. This study introduces a novel framework that defines the spacetime metric and its inverse via a set of underlying vector fields. By expressing the metric tensor in terms of these fields, we derive the corresponding geodesic and Einstein field equations, revealing striking structural parallels with electrodynamics—most notably, the natural emergence of a covariant Lorentz-like force and a stress-energy tensor mathematically analogous to that of electromagnetic fields. To demonstrate the astrophysical viability of this approach, we solve the equations for a spherically symmetric, static mass distribution, yielding a non-singular alternative to the Schwarzschild metric. The practical validity of this metric is subsequently tested through the recalculation of the perihelion precession of planets and the gravitational deflection of light. The results provide precise agreement with perihelion precession observations while predicting a slightly modified value for light deflection (1.90 arcsec), offering a potential theoretical basis for the historical discrepancies observed between visible light and radio wave measurements during solar eclipses.