Connection forms associatedwith the Levi-Civita connection

Question

Say we have a Riemannian manifold $(M,g)$ of dimension $3$ with an orthonormal frame $\{e_i\}_{i=1}^3$. Denote by $\theta^i$ the associated dual basis, that is, $\theta^i(e_j)=\delta_i^j$ for every $i,j=1,2,3$. Then, the connection $1$-forms $\omega_j^i$ associated with the Levi-Civita connection $\nabla$ will satisfy the equations $$d\theta^i=-\sum_{j=1}^3 \omega_j^i \wedge \theta^j,$$ and $$\nabla_{e_i}e_j=\sum_{k=1}^3 \omega_j^k(e_i)e_k.$$ My question is if the $\omega_j^i$ will span the cotangent space $T^*M$. I have used this property to actually compute $\omega_j^i$ in some excersises, and it always is the case that the $\omega_j^i$ span $T^*M$, but I don't know if this is true.