Higher-dimensional reflections

Question

I would like to know if it is possible to find a hierarchy on reflections in the following sense:

Let $V$ be a finite-dimensional euclidean vector-space with standard inner product $\langle\, \_ \,,\, \_ \, \rangle$.

1. a 1-dimensional reflection should be the map $s:V \to V$, $v \mapsto -v$
2. a 2-dimensional reflection should be the map $V \to \text{Aut}(V)$, $v \mapsto \left(s_v: w \mapsto w-2 \frac{\langle v,w \rangle}{\langle v,v \rangle}v \right)$
3. I'm already not sure, what a 3-dimensional reflection should be, but I would guess that the involved involution(s) $s$ now has(have) two indices.
4. ...

Clearly, $s_v(v)=s(v)$, so I believe that $n$-dimensional reflection should be special cases of $n+1$-dimensional reflections

Sorry for this very vague question, I guess I'm counting on this being a well-known structure (if it exists) that is immediately recognized by someone who knows it..

EDIT: To avoid misunderstanding, $n$-dimensional reflections don't necessarily live in $n$-dimensional vector spaces. By an $n$-dimensional reflection I don't mean reflection through the hyperplane orthogonal to some vector, except for $n=2$.

Answer 1

In fact what you describe for dimension $2$, i.e. the map $V \to \text{Aut}(V)$, $v \mapsto \left(s_v: w \mapsto w-2 \frac{\langle v,w \rangle}{\langle v,v \rangle}v \right)$ is the general way to describe a reflection in any dimension.

You'll see that taking $v=1$ gives you what you describe for dimension equal to $1$.

And see this wikipedia article for higher dimensions.