The perpendicular hyperplane is the *standard* way.
The gradient of a differentiable function $f: \Bbb R^{n+1} \rightarrow \Bbb R$ gives another idea. Select one direction to be the height $h$. Then, given a vector $v$, take the hyperplane whose inclination of all the other directions against the height is given by $v$.
Are there other (describable) ways?