I am learning Cartan's moving frame from chapter 6 of S.S. Chern's book "Lectures on differential geometry" ([Google books][1]). Suppose the moving frame in $E^N$ is denoted by $(p;e_1,\cdots,e_N)$, then we can apply an infinitesimal motion on $(p;e_1,\cdots,e_N)$ to get the equation (3.22) on p.202 Chern's book:  
\begin{equation}
\left\{ \begin{split}
dp=\displaystyle{\sum_{\alpha=1}^{N}\omega^{\alpha}e_{\alpha}}\\
de_{\alpha}=\displaystyle{\sum_{\beta=1}^{N}\omega^{\beta}_{\alpha}e_{\beta}}
\end{split} \right.
\end{equation} 
where the $\omega^{\alpha}$ and $\omega^{\beta}_{\alpha}$ are one forms. In my understanding, the $e_{\alpha}$'s  are understood as $\mathbb{R}^N$ valued **functions** (of some $N$ variables $(u_1,\cdots,u_N)$) since we can do the operator $d$ to them. In other words, the $e_{\alpha}$'s **are not really vector fields** in $\mathbb{R}^N$ (by vector field I mean a **contravariant 1-tensor field**) since we can't do the $d$ operator to a vector field. However, Chern's book treat the $e_{\alpha}$'s as vector fields and it takes the one forms $\omega^{\beta}_{\alpha}$ as connection 1-form as the Levi-Civita connection for $\mathbb{R}^N$ and do covariant derivative $D$ to $e_{\alpha}$ (this is (3.38) on p.207 Chern's book):
$$
De_{\alpha}=\omega_{\alpha}^{\beta}e_{\beta}
$$
(Here, I take $m=N$ in Chern's book). So my question is why we can treat the $e_{\alpha}$ as contravariant fileds in $\mathbb{R}^N$ and do the $d$ operator to them? Why the one forms $\omega_{\alpha}^{\beta}$ (gotten from the equation for the moving frame) are exactly the **connection 1-forms** for the Levi-Civita connection on $\mathbb{R}^N$?

  [1]: https://books.google.ca/books?id=Mvk7DQAAQBAJ&printsec=frontcover&hl=zh-CN#v=onepage&q&f=false