Two parameters are introduced that uniquely characterize the state of a third-order symmetric tensor. We show that the proposed parameters arise from the uniform metric in the matrix space; thus the joint PDF of these parameters can be used to determine the geometrical statistics of any third-order symmetric tensor. We use this joint PDF to describe the states of the subgrid-scale stress, which is of central interest in large-eddy simulation. Direct numerical simulation of forced isotropic turbulence is used in our a priori tests. With the proposed parameterization we can also assess the most probable flow configuration at the scales of motion just above the Kolmogorov scale. We test four different subgrid-scale models in terms of how well they predict the structure, or state, of the subgrid-scale stress. It is found that models based on truncated Taylor series do not produce an adequate distribution of states, even if augmented by a turbulent viscosity term. On the other hand, models based on the scale-similarity assumption predict a distribution of states that is close to actual.