Mariano's answer should be enough, if you read your link to Wikipedia carefully, but maybe I could add some hints.
If you want to do a reflection about a hyperplane containing the origin, you pick a unitary orthogonal vector to that hyperplane, $v$, and then write the vector $x$ you want to reflect about $[v]^\bot$ as a sum
$$
x = \lambda v + u
$$
with $u \in [v]^\bot$. Now, you want to determine $\lambda \in \mathbb{R}$ in order this to be true:
$$
v \cdot x = \lambda v\cdot v + v\cdot u = \lambda \ .
$$
So, necessarily
$$
x = (v\cdot x) v + u \ .
$$
Hence, if you want to reflect $x$ about $[v]^\bot$, you just substract two times its $v$-component:
$$
P_v x = x - 2 (v\cdot x) v = x - 2 v (v^t x) = (I - 2vv^t ) x \ .
$$
So any (orthogonal) reflexion about any hyperplane $[v]^\bot$ is of the form
$$
P_v = I - 2vv^t \ .
$$
That is, a Housholder reflection.