I am reading some notes about Riemannian Structures. In definition of moving frame I see blow text and can't understand what $dx$ is:
By a moving frame in $U\subseteq \mathbb{R}^N$ we mean a choice of smoothly varying orthonormal bases $\{e_1(x),\ldots, e_N(x)\}$ for all $\mathcal{T}_xU, x\in U$. Taking exterior derivatives we obtain $$dx = \sum_A \omega_A e_A , de_A=\sum_B \omega_{BA}e_B$$ where $\omega_A$'s and $\omega_{BA}$'s are 1-forms.
Think of $x\colon U\hookrightarrow\mathbb R^n$. Then you can think of $dx$ both as an $\mathbb R^n$-valued $1$-form on $U$ and as a fancy way of thinking of the identity map on $T_xU\cong\mathbb R^n$ (as a tensor field of type $(1,1)$ on $U$).
Note that $\{e_A\}$ and $\{\omega_A\}$ are dual bases. We are writing, more pedantically, the identity map as $$dx=\sum_A \omega_A\otimes e_A, \quad\text{i.e., for any }v\in T_xU,\quad v=dx(v)=\sum_A \omega_A(v)e_A\,.$$