I was looking for an explicit formula for the codifferential of a 2-form $\omega=\omega_{ij}dx^i\wedge dx^j$ but I found two expressions that look different. One from Jost's book *Riemannian Geometry and Geometric Analysis* according to which
$$
(d^{*}\omega)_{j}=-g^{kl}\left(\partial_{l}\omega_{kj}-\Gamma_{kl}^{s}\omega_{sj}\right)
$$
and the other here in Stackexchange reading
$$
(d^{*}\omega)_{j}=-g^{kl}\nabla_k\omega_{lj}
$$
Being $\nabla_k$ the covariant derivative. Explicitly
$$
(d^{*}\omega)_{j}=-g^{kl}\left(\partial_{l}\omega_{kj}-\Gamma_{kl}^{s}\omega_{sj}-\Gamma^s_{kj}\omega_{ls}\right)
$$
I tried but I can't tell which of the two formulas is correct. Can anybody point out a reference where the validity of the correct expression is proved?

Thanks in advance.