Surfaces in $\mathbb{R}^n$ locally isometric to $S^{n-1}$

Question

As a consequence of the Theorema Egregium we know that $\mathbb{R}^2$ is not even locally isometric to $S^2$. It's then easy to generalise this so that $2$ is any $n\geq 2$. Now, it's easy to come up with surfaces in $\mathbb{R}^3$ that are "legitimately" distinct from $\mathbb{R}^2$, but locally isometric to it, like a cylinder. However, I cannot come up with a single example of a surface in $\mathbb{R}^n$ that is locally isometric to an $(n-1)$-sphere of radius $r$, and is not merely a subset of a $(n-1)$-sphere of radius $r$. I was wondering if any such surface actually exists.