I am trying to learn the vielbein formalism and have a question for the example of the Riemann sphere $S^2$. I am afraid my question is rather elementary, as it seems to be a simple sign error. Still, could someone help me figure this out?

On the sphere with coordinates $(x,y,z) = (\cos ϕ \sin θ, \sin ϕ \sin θ, \cos θ)$ and metric $ds^2 = dθ^2 + \sin θ^2 dϕ^2$, we can define the zweibein

$$ e_θ = ∂_θ , \quad e_ϕ = \frac{1}{\sin θ} ∂_ϕ$$

The Levi-Civita connection for the metric is torsion-free, which means

$$ \nabla_{e_ϕ} e_θ - \nabla_{e_θ} e_ϕ = [e_ϕ,e_θ] $$

A separate calculation shows that $\nabla_{e_θ}e_ϕ = 0$, so we can use this formula to quickly calculate the connection form $ω_{ab}$:

$$ \nabla_{e_ϕ} e_θ = [e_ϕ,e_θ] = -∂_θ(\frac{1}{\sin θ})·∂_ϕ = \cot θ · e_ϕ \equiv ω_{θ\ \ }^{\ \ ϕ}(e_ϕ) e_ϕ$$

Unfortunately, this calculation seems to be wrong, because it contradicts the statement

$$ ω^{ϕ\ \ }_{\ \ θ}(e_ϕ) = \cot θ $$

that I found in some lecture notes (formula (2.345)). The connection form is antisymmetric, so one of the two values should be $-\cot θ$, but I can't decide which.

Can somebody help me find the source of this sign discrepancy?