How to define a 'surface measure' on a smooth boundary

Question

Let $U$ be a bounded open set in $\mathbb{R}^n$ with a $C^1$ boundary $\partial U$.

Then I encounter expressions like $L^p(\partial U)$. I know how to define the surface measure for $S^{n-1}$; but for an arbitrary $C^1$ boundary, how can one define a measure? Could anyone help me?