Trivializations which preserve orthogonal structures I am going to quote a passage from Bott and Tu and ask some questions:

We now show that the structure group of every real vector bundle $E$ may be reduced to the orthogonal group. First, we can endow $E$ with a Riemannian structure as follows: Let $\{U_\alpha \}$ be an open cover of $M$ which trivializes $E$. On each $U_\alpha$, choose a frame for $E|_{U_\alpha}$ and declare it to be orthonormal. Let $<,>_\alpha$ denote this inner product on $E|_{U_\alpha}$. Now use a partition of unity to splice them together. This will be an inner product over all of $M$. As a trivialization of $E$, we take only those maps $\phi_\alpha$ that send orthonormal frames of $E$ relative to this metric to orthonormal frame of $R^n$. Then the transition functions preserve orthonormal frames....

What do they mean by "declare"?  How do we know we can always "take only those maps $\phi_\alpha$ that send orthonormal frames of $E$ relative to this metric to orthonormal frame of $R^n$."? ${}$${}$${}$
 A: Let $\{e^1_\alpha, \cdots, e^n_\alpha\}$ be a frame in the trivialization $U_\alpha$, by declaring it to be orthornormal, it means that you define a metric $\langle \cdot, \cdot\rangle_\alpha$ on each fiber $E_x$, where $x\in U_\alpha$ by $\langle e^i_\alpha, e^j_\alpha\rangle = \delta_{ij}$. 
For the second question, assume we already has a metric $\langle \cdot, \cdot\rangle$ defined on $E$. Let $\{U_\alpha\}$ be any local trivialization of $E$. These are mapping 
$$ \psi_\alpha : \pi^{-1}(U_\alpha) \to U_\alpha \times \mathbb R^n$$
which is linear on each fiber. Let $\{ e^1_\alpha, \cdots, e^n_\alpha\}$ be an local orthonormal basis on $E|_{U_\alpha}$. Then define 
$$L : U_\alpha \times \mathbb R^n \to U_\alpha \times \mathbb R^n,\ \ \  L(x, \phi(e^i_\alpha)_x) = (x, \delta_i)$$ 
where $\{\delta_1, \cdots, \delta_n\}$ is the standard basis on $\mathbb R^n$. Thus $\phi = L\circ \psi$ send orthonormal basis of $E$ relative to $\langle\cdot, \cdot\rangle$ to orthornomal basis of $\mathbb R^n$. 
