I am reading Milnor's "Characterstic classes". I am currently working through the chapter on Chern classes. In particular, he shows that any complex vector bundle induces a canonical orientation on it's underlying real vector bundle. At the end of the proof of this statement (which is on the first page of the chapter) he states
"Now, if $\omega$ is a complex vector bundle, then applying this construction to every fiber of $\omega$, we obtain the required orientation for $\omega_{R}$."
Where $\omega_{R}$ is the underlying real vector bundle.
I am having a hard time understanding why, by orienting the fibers of the underlying real vector space in this way (I review the construction below), one will always be able to orient the fibers in a "continuous" way across the whole manifold, thereby providing an orientation of the entire bundle.
Obviously it can be done on any given elementary neighborhood, but I worry that the manifold may be too twisted and this construction will eventually run into a contradiction. If someone could help me to understand why this construction can always be done consistently I would appreciate that greatly.
As a reminder, the construction is as follows:
Given a complex vector space of dimension $n$, we choose a complex basis $\{a_{1},\ldots,a_{n}\}$ and define the orientation of the underlying real vector space using the ordered $2n$-tuple $\{a_{1},i a_{1},\ldots,a_{n},i a_{n}\}$ as our basis. This is well defined since every complex basis for the space can be continuously connected by a path in $GL(n,\mathbb{C})$ (since this group is connected) and a continuous transformation cannot change orientation.