Assume we have a function $\cases{\phantom{f(}{\bf x}\phantom) \in \mathbb R^N\\f({\bf x})\in \mathbb R^1 }$
Furthermore assume it is differentiable so we can calculate a gradient $\nabla f \in \mathbb R^N$
Now, how can we build an ON-basis for a tangent-plane $$\bf T = [t_1,t_2,\cdots,t_{N-1}]$$ in such a way that for any $\bf x$, every basis vector $\bf t_k(x)$ will vary smoothly with $\bf x$? We know that $${\bf T}^T(\nabla f)({\bf x})={\bf 0}$$
Own work: In 2 dimensions ($N=2$), this is easy : we can just choose the typical $$\begin{bmatrix}0&-1\\1&0\end{bmatrix} (\nabla f)({\bf x}) = \begin{bmatrix}-(\nabla f)({\bf x})_y\\(\nabla f)({\bf x})_x\end{bmatrix}$$ Which will be independent of anything else.
I was thinking we can define some kind of linear orthogonal operator which is a function of $\nabla f$ or it's derivatives in some sense, and is working on $\bf T$ or it's transpose. But I can't seem to formulate anything useful.