I was pondering angles in spaces of dimension greater than $2$, and ended up coming up with the following:
Definition 0. Let $V$ denote a real vector space. Then a ray in $V$ is a subset $\alpha$ of $V$ such that for all $x,y \in \alpha$, there exists unique $a \in \mathbb{R}_{>0}$ such that $ax = y$.
Definition 1. Let $V$ denote a real vector space equipped with a norm and an orientation. Then an angle in $V$ is a bijective orientation preserving isometry $V \rightarrow V$, and the angle from a ray $\alpha$ to a ray $\beta$ is defined as the unique angle $\theta$ such that firstly, $\theta(\alpha) = \beta$, and secondly $\theta$ stabilizes the orthogonal complement of $\mathrm{span}\{\alpha,\beta\}$.
Note that if $V$ is two-dimensional, we can get a bijective correspondence between angles and rays by choosing a ray to be the "identity angle", and these can in turn be put into bijective correspondence with the quotient group $\mathbb{R}/2\pi\mathbb{Z}$. Anyway, I'm just wondering what "angles" in the sense that I use the word above are really called, and how best to refer to the "angle between" two rays.
This definition means that the nonnegative reals don't form a ray, and the bisector of an angle $CAB$ doesn't contain $A$.
That's OK, I guess, but the real problem is that the set of positive rationals forms a ray, and that just seems weird.
When you say "stabilizes", do you mean pointwise or setwise? For "setwise", in 4-space you can send $x$ to $y$ and $y$ to $-x$, but send $z \to -z, w \to -w$, and still stabilize the $zw$-plane, but without fixing it. That kinda kills "uniqueness", since sending $z\to z, w \to w$ also provides an "angle." If you mean pointwise, then your definition can be turned into something equivalent:
Pick a basis $u_1, \ldots, u_{n-2}$ for the orthogonal complement, and extend it to an oriented basis for $n$-space by adding in the vector $x$ and the vector $z$. Then the rotation you've described splits as a sum of an $n-2$ dimensional identity and a 2-dimensional rotation in $xz$ coordinates. And then the "angle" is just the ordinary $2D$ angle measure. I guess I'm saying that once you realize that all the action is happening in a 2-plane, the complementary $n-2$ plane is just kind of a red herring.
What I like about your definition is that it emphasizes transformation.
What I don't like is that one of the useful things about angles is angle measure, and it's a little tricky to get that angle measure from the transformation.
To answer your last question: the angle between two rays is usually called "the angle between two rays", where a ray consists of all nonnegative multiples of some nonzero vector, and where the angle is measured by taking nonzero vectors $u$ and $v$ in each ray and defining $$ \theta = \cos^{-1} \left( \frac{u \cdot v}{\|u\|\|v\|}\right). $$