Let $S$ be the unit sphere in $\mathbb{R}^3$. Is it possible to embed $S$ isometrically into some $\mathbb{R}^n$, $n>3$, such that the image does not lie on any 3-dimensional affine subspace?

More generally, if $S$ is a compact surface with positive Gaussian curvature, we know it can be isometrically embedded into $\mathbb{R}^3$ uniquely, can there be a non-trivial isometric embedding into higher dimensional Euclidean space? Thanks!

Added May 9:

After seeing the comments I realized the question is trivial. While it looks like a HW question, I'm actually thinking about a research problem and thought that a unique isometric embedding may help, so I set up this as a toy problem (without realising that $\mathbb R^3$ has a simple but non-trivial embedding into $\mathbb R^4$). Anyway, although not related to my problem in mind, I am curious to know if the same is true for $\mathbb {H}^n$ or $\mathbb S^n$? I don't see how e.g. $\mathbb S^n$ can be non-trivially into $\mathbb S^{n+1}$. It seems that for such an embedding to exist, all except one of the principal curvatures vanish. I don't know how to construct such a submanifold globally.