Expressing affine connection in tetrad formalism

[lapo]

Hi, some one know the expression of the affine connection Γ in terms of tetrad formalism? I would like also some references if it's possible, i found a hit but i think that is wrong... please help me it's so important!

 

[PWiz]

$$Γ^{\lambda} _{μν} = \frac{1}{2} (e^κ ⋅ e^λ) ( ∂_μ (e_κ ⋅ e_ν) + ∂_ν (e_κ⋅e_μ) - ∂_κ (e_μ⋅e_ν))$$

 

[lapo]

$$Γ^{\lambda} _{μν} = \frac{1}{2} (e^κ ⋅ e^λ) ( ∂_μ (e_κ ⋅ e_ν) + ∂_ν (e_κ⋅e_μ) - ∂_κ (e_μ⋅e_ν))$$

I'm loking for this relation:
\begin{equation}
\Gamma^c_{ab}=-\Omega_{ab}\,^c+\Omega_b\,^c\,_a-\Omega^c\,_{ab} \end{equation}
where $\Omega$ is:
\begin{equation}
\Omega_{ab}\,^c=e^\mu\,_ae^\nu\,_b\partial_{[\mu}e_{\nu]}\,^c
\end{equation}
But i have an extra term and i don't understand where is the mistake