Find the Gauss map for $2$-sheeted hyperboloid $$x_1^2-x_2^2-\dots-x_{n+1}^2=4$$ , $x_1\gt0$
I know that the orientation should be taken as $\bar{N}(p)=-\frac{\nabla f}{||\nabla f(p)||}$ where $p\in$ , $S$ is an n-surface. Then $\bar{N}(p)=(p,N(p))$ I take $p=(x_1,x_2,...,x_{n+1})$ . I need to find the map $\bar{N}: S \to S^n$.