I have a seemingly innocent (possibly trivial) question about the notion of orientation on vector spaces.
Given a three-dimensional subspace $\Sigma$ of $\mathbb{R}^{d \geq 4}$ (the latter equipped with its standard orientation), let $v_{1}$ and $v_{2}$ be two orthogonal vectors in $\Sigma$.
Question. Can I use the "right hand rule" to define a "positively oriented" basis of $\Sigma$? In other words, if for some reason $v_{1}$ and $v_{2}$ have a natural order, is there any obstruction in defining (canonically) a positively oriented basis of $\Sigma$?
I may misunderstand the problem...
Anyway, from what I read, $\Sigma$ is a 3D space, so you can equip it with a canonical basis $\{e_1,e_2,e_3\}$, which provides an orientation of $\Sigma$, and then work in $\Sigma$ as you would do in the usual 3D oriented space, using the right-hand rule.
In other words, from what you described, I don't see what the role of $\mathbb R^{\ge 4}$ is.