In this work, we show an extension of the mixed Finite Element Method (FEM) from small strain elasticity to large strain problems. This work develops a finite element formulation that independently approximates four primary fields: the stresses, logarithmic stretches, rotations vector, and displacements. Each field is associated with a set of equations namely, conservation of linear momentum, conservation of angular momentum, constitutive (physical) equation, and consistency equation between displacements and deformation. Unlike traditional FEM formulations, the stresses are approximated in H(div) space, while the remaining three fields are approximated in L^2 space. The rotations vector and rotation tensor are connected by the typical exponential map. This formulation results in a very sparse system of equations that is easy to parallelise, enabling highly-scalable and robust solvers. To further enhance computational efficiency, a hybridisation technique is used for the displacement and stress fields, significantly reducing the size of the global stiffness matrix. In addition, in order to ensure a robust algorithm, a modified neo-Hookean material model expressed in terms of logarithm of stretches, is used to overcome the pathologies associated with the inverse of the deformation gradient term in the constitutive law. This novel finite element technology is implemented in open-source finite element library, MoFEM, and will be shown to tackle complex problems such as nearly incompressible soft materials through several numerical benchmark problems. The mixed finite element formulation opens new possibilities to tackle robust problems in Data-Driven approaches for large strains and multi-field formulations for computational plasticity and efficient error estimators for p-adaptivity.