Can we extend a function on surface to a function on $\mathbb{R}^3$

Question

I'm studying differential geometry right now. I've encountered a problem considering the definition of a function/differential forms on a surface $S \in \mathbb{R}^3$.

Suppose we're given a $C^1$ function $f$ on $S$, can we say that there must be a restriction of some function $F$ on $\mathbb{R}^3$ such that $F|_{S} = f$. And, does this result hold for differential forms as well?