I'm trying to prove this exercise from G&P book, but I don't know if I'm right in my sketch: here it follows
By the smooth Jordan--Brouwer Separation Theorem, $\mathbb{R}^n \setminus \Sigma$ has exactly two connected components: a bounded interior $V$ and an unbounded exterior $U$ then we define a unit normal vector field $N$ on $\Sigma$ taking for every $x\in \Sigma$, the unit vector normal to $T_x\Sigma$ pointing into $U$ which is well-defined and smooth because $\Sigma$ is smooth and the choice of exterior is consistent. Then given $N$, we define an orientation on $T_x\Sigma$ saying that a basis $v_1,\ldots,v_{n-1}$ of $T_x\Sigma$ to be positively oriented iff $(N(x),v_1,\ldots,v_{n-1})$ is an oriented basis for $\mathbb{R}^n$. This orientation varies smoothly with $x$, hence $\Sigma$ is orientable.
