We study entanglement of electron spins in many-body systems based on the Green's function approach. As an application we obtain the two-particle density matrix of a non-interacting electron gas and identify its two-spin density matrix as a Werner state. We calculate entanglement measures, a classical correlation, mutual information, and a pair distribution function of two electrons at zero and finite temperatures. We find that changes of entanglement measures are proportional to $T^2$ at low temperatures.
