Cartan's first structural equation

(@lee2006riemannian exercise 4-5 page 64)
Let $\nabla$ be a linear connection on $M$ , let ${e_{i}}$ be a local frame on some open subset $U \subset M$ , and let ${ω^{i}}$ be the dual coframe. Denote by $T_{i j}^{k}$ the structure coefficients.
We know that there is a uniquely determined matrix of 1-forms $Θ_{j}^{i}$ on $U$ , called the connection 1-forms for this frame, such that

\nabla_{X} e_{i} = X ⌟ Θ_{i}^{j} e_{j}

for all $X \in T M$ .

Theorem (Cartan’s first structure equation)

d ω^{k} = ω^{i} \land Θ_{i}^{k} + τ^{k},

where ${τ^{1}, . . ., τ^{n}}$ are the torsion 2-forms, defined in terms of the torsion and the frame ${e_{i}}$ by

τ (X, Y) = τ^{j} (X, Y) e_{j} .

$◼$
Proof
We take $e_{1}, e_{2}$ without lost of generality, and observe that the torsion satisfies

τ (e_{1}, e_{2}) = \nabla_{e_{1}} e_{2} - \nabla_{e_{2}} e_{1} - [e_{1}, e_{2}]

The $k$ th component will be

τ^{k} (e_{1}, e_{2}) = e_{1} ⌟ Θ_{2}^{k} - e_{2} ⌟ Θ_{1}^{k} + T_{12}^{k}

But also $d ω^{k} (e_{1}, e_{2}) = T_{12}^{k}$ , since $d ω^{k} = \sum_{i < j} T_{i j}^{k} ω^{i} \land ω^{j}$ . It only rests to show that

ω^{i} \land Θ_{i}^{k} (e_{1}, e_{2}) = e_{2} ⌟ Θ_{1}^{k} - e_{1} ⌟ Θ_{2}^{k}

and this is true, since

ω^{i} \land Θ_{i}^{k} (e_{1}, e_{2}) = \sum_{i} det (\begin{matrix} e_{1} ⌟ ω^{i} & e_{1} ⌟ Θ_{i}^{k} \\ e_{2} ⌟ ω^{i} & e_{2} ⌟ Θ_{i}^{k} \end{matrix}) = e_{2} ⌟ Θ_{i}^{k} - e_{1} ⌟ Θ_{i}^{k} .

$◼$

Comment: I think that Cartan's first structural equation is used sometimes to recover a connection expressed in a particular frame/coframe if we know the torsion. Mainly with the Levi-Civita connection of a Riemannian metric and orthonormal frames
In general, you need to know more information about the "relation" of the chosen frame with the connection. But in the particular case in which the frame is orthonormal, now this is enough data to recover the metric using the torsion (null torsion, for example). But anyway, if you assume orthornormality you can compute the metric, compute Christoffel symbols and then change to the desired frame, not needing Cartan's first structure equations... I guess they are used as a shortcut, but they are not indeed needed...