Let $\{E_1,...E_n\}$ be a set of vector fields defined on a domain $U \subset \mathbb{R}^n$ such that for all $p \in U$, $\{E_1(p),...,E_n(p)\}$ is an orthonormal basis for $T_pU$ (tangent space at $p \in U$) in the sense that $$E_i(p) \cdot E_j(p) = \delta_{ij}$$ (dot product on $\mathbb{R}^n$) Such a set is an orthonormal frame relative to the standard inner product. If we take $T_pU = \mathbb{R}^n$, we can let each $E_i$ be a differentiable map $E_i:U \rightarrow \mathbb{R}^n$, and we can consider the tangent map $(E_i)_*:TU \rightarrow T\mathbb{R}^n$. For $i,j = 1,...,n$ define for all vector fields $V$ on $U$, $$\omega_{ij}(V) = (E_i)_*(V)\cdot E_j$$ We want to show that $\omega_{ij}$ is a differential form $\forall\:\: i,j$. The hint is to show that it is smoothly varying and linear on each tangent space. I have no idea how this can be done. Any help is much appreciated.
We are also asked to show that $\omega{ji} = -\omega{ij}$ by differentiating $E_i \cdot E_j = \delta{ij}$. How in the world do we do this? I feel like my advanced calc is useless in showing me a way forward.
\begin{align*} 0 = d(E_i\cdot E_j) &= dE_i\cdot E_j + E_i\cdot dE_j = \sum \omega_{ik}E_k\cdot E_j + E_i\cdot \sum\omega_{jk}E_k \\ &= \sum \omega_{ik}\delta_{kj} + \sum\omega_{jk}\delta_{ik} = \omega_{ij}+\omega_{ji}. \end{align*}