Fixing orientation by fixing a frame in a tangent space

[Unconscious]

I would like to show that fixing the orientation of k-manifold smooth connected $S$ in $\mathbb {R} ^ n$ is equivalent to fixing a frame for one of its tangent spaces.

What I know is that different orientations correspond to orienting atlases containing maps that cannot be consistent with maps in other orienting atlases of other equivalence classes, when their domains of action overlap.

How could we pass from this fact, to the fact that it is enough to fix a frame anywhere $x_0 \in S$ to determine the orientation of $S$?

 

[quasar987]

Well this is certainly not true as written since any connected manifold, orientable or not, can be embedded in R^n for n large enough. You probably meant simply connected. Look at Claudio Gorodski's answer here: https://math.stackexchange.com/questions/786483/simply-connected-manifolds-are-orientable