For two majorana field $\psi$ and $\chi$, which satisfy $\psi_{c}=\psi$ and $\chi_{c} = \chi$, where the charge-conjugation operation we define as $$ \Psi_{c} = C \Psi^{\ast} $$ where we work in the Weyl basis so that the matrix $C$ is given by $C = -i\gamma^2$.
It seems to me that the above implies $\bar{\psi}_{c} = ( C \psi^{\ast} )^{\dagger} \gamma^0 = \psi^{T} C^{\dagger} \gamma^0 = - \psi^{T} \gamma^0 C$, and so from this it follows that: $$ \bar{\psi} \chi = \psi^{\dagger} \gamma^0 \chi = \chi^{T} \gamma^0 \psi^{\ast} = \chi^{T} \gamma^0 C C \psi^{\ast} = - \left( - \chi^{T} \gamma^0 C \right) ( C \psi^{\ast} ) = - \bar{\chi}_c \psi_c = - \bar{\chi}\psi $$ where we have used $CC=I$, and the last equality uses the fact that these fields are Majorana fermions.
This implies that $\bar{\psi} \chi + \bar{\chi} \psi =0$ which seems wrong....where am I making a mistake?