A new locking-free formulation of a three-dimensional shear-deformable beam with large deformations and large rotations is developed. The position of the centroid line of the beam is integrated from its slope that is related to the rotation of a corresponding cross section and stretch and shear strains. The rotation is parameterized by a rotation vector, which has a clear and intuitive physical meaning. Taylor polynomials are used for certain terms that have zero denominators to avoid singularity in numerical implementation. Since the rotation vector can have singular points when its norm equals $2m\pi $, where $m$ is a nonzero integer, a rescaling strategy is adopted to resolve the singularity problem when there is only one singular point at a time instant, which is the case for most engineering applications. Governing equations of the beam are obtained using Lagrange's equations for systems with constraints, and several benchmark problems are simulated to show the performance of the current formulation. Results show that the current formulation does not suffer from shear and Poisson locking problems that the absolute nodal coordinate formulation (ANCF) can have. Results from the current formulation for a planar static case are compared with its exact solutions, and they are in excellent agreement with each other, which verifies accuracy of the current formulation. Results from the current formulation are compared with those from commercial software abaqus and recurdyn, and they are in good agreement with each other; the current formulation uses much fewer numbers of elements to yield converged results.