In a Zoom lecture given by a mathematical physics professor, if I recalled correctly, he explained that the in 1+1 dimensional spacetime (or 2 dimensions in short), the "action" of fermions (spinors) has an anti-symmetric Dirac operator.

1. Say, if the $\psi$ is a Dirac spinor, he wrote down an action $$ \int d^2x \sqrt{g} \bar{\psi} (i \gamma^\mu D_\mu) \psi $$ and (I think) he claims that the operator $(i \gamma^\mu D_\mu)$ is an anti-symmetric matrix.

2. Say, if the $\chi$ is a Majorana spinor, he wrote down an action $$ \int d^2x \sqrt{g} {\chi} (i \gamma^\mu D_\mu) \chi $$ and (I think) he claims that the operator $(i \gamma^\mu D_\mu)$ is also an anti-symmetric matrix.

Is this true that the anti-symmetric matrix has something to do with these fermions (spinors)? or fermion statistics? Why?

p.s. Maybe the first case the $D$ operator is complex, and the second case that the $D$ operator is real (?).