We develop a mixed finite element method for the coupled problem arsing in the interaction between a free fluid governed by the Stokes equations and flow in poroelastic medium modeled by the Biot system. Mass conservation, balance of stress, and the Beavers-Joseph-Saffman condition are imposed on the interface. We consider a fully mixed Biot formulation based on a weakly symmetric stress-displacement-rotation elasticity system and Darcy velocity-pressure flow formulation. The interface conditions are incorporated through the introduction of the traces of structure velocity and Darcy pressure as Lagrange multipliers. Existence and uniqueness of a solution are established for the continuous weak formulation. Stability and error analysis is performed for the semi-discrete continuous-in-time mixed finite element scheme. Numerical experiments are presented in confirmation of the theoretical results.