Asymptotic of the measure of a covering tube of a submanifold of dimension $n-2$

Question

Let $M$ be a bounded $C^1$ submanifold of $\mathbb{R}^n$ of dimension $n-2$ or less. Consider the tube $T(p)$ of all points $x$ in $\mathbb{R}^n$ such that $d(x,M)<1/p$, for all $p\in \mathbb{N}$ (where $d$ is a distance associated to a norm on $\mathbb{R}^n$).

Do we have $\lambda(T(p))=\mathcal{O}_{p\to \infty}(1/p^2)$ ? (where $\lambda$ is the Lebesgue measure on $\mathbb{R}^n$.)