Let $d\in\mathbb N$, $k\in\{1,\ldots,d\}$, $f:\mathbb R^d\to\mathbb R^k$ be differentiable, $a\in\mathbb R^k$ be a regular value of $f$ and $M:=\{f=a\}$.
We know that $M$ is a $(d-k)$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$ and $T_x\:M={\operatorname{span}\left\{\nabla f_1(x),\ldots,\nabla f_k(x)\right\}}^\perp$ for all $x\in M$.
Let $x\in M$. We can immediately conclude that $\nabla f_1(x),\ldots,\nabla f_k(x)$ is a basis of $N_x\:M=(T_x\:M)^\perp$.
But is there also an explicit form for a (orthonormal) basis of $T_x\:M$ (which is a $(d-k)$-dimensional space)?