One of the definitions of the dot product of two vectors is the following

$$\vec{a} \bullet \vec{b}\ \ \stackrel{\text{def}}{=} \ \ \|\vec{a}\| \,\|\vec{b}\|\cos(\theta)$$

Where $\theta$ denotes the angle between the vectors $\vec{a}$ and $\vec{b}$. But for that angle $\theta$ to 'make sense', both $\vec{a}$ and $\vec{b}$ must lie in the same plane correct? i.e. There must exist some two-dimensional space, e.g. $\mathbb{R^2}$, that both $\vec{a}$ and $\vec{b}$ are an element of.

Now we can *always* find a plane that both $\vec{a}$ and $\vec{b}$ fall in. But now the question arises, are we finding the angle $\theta$ with respect to the plane that they lie in (what I'm trying to say is: Is $\theta$ denoting the angle between $\vec{a}$ and $\vec{b}$ in a plane that they both fall in?), or is it with respect to the basis vectors of of the original vector space that they lie in initially.

Here's an example that will hopefully get across what I'm asking:

Take $\vec{a}, \vec{b} \in \mathbb{R^4}$.

If we can find a plane that both $\vec{a}$ and $\vec{b}$ both lie in, then *'the angle between them'*, $\theta$, makes sense. If not then what essentially is being said is $\theta$ is the angle between two vectors in four-dimensional space, and I'm not sure that an angle in $\mathbb{R^4}$ has any sort of meaning.

My question then essentially boils down to the following. 

* Are angles only defined (or do they only have meaning) in $\mathbb{R^2}$?
* If we have two vectors in $\mathbb{R^n}$, is the only way to find the 'angle between them' by first finding a plane (a two-dimensional vector space) that they both lie in, and then solving for the angle with respect to that plane?