I see somewhere that we can add a Majorana mass term, $m\psi^Ti\sigma^2\psi+h.c.$, to a Weyl fermion Lagrangian, where $\psi$ is a two-component spinor. However, it seems that this term simply vanishes due to the anti-symmetry of $i\sigma^2$. Did I miss anything here?
I found the answer in one of the problems in Peskin.
Because in the corresponding quantum theory, this spinor field will be anti-commuting variables, therefore it makes sense to start with classical anti-commuting Grassmann numbers for the classical field. In this sense, the mass term will not vanish.