There is the following well known result about manifolds in $\mathbb{R^n}$:
Theorem: Let $M\subset\mathbb{R^n}$ be a surface (manifold) of class $C^k$ $k\geq 1$ and dimension $m$. If there are $n-m$ continuous normal ($v_{i}(p)$ belongs to the orthogonal complement of $T_{p}M$ for all $p\in M$ and $i$) vector fields $v_1,\dots,v_{n-m}:M\to\mathbb{R^n}$ such that $v_1(p),\dots,v_{n-m}(p)$ are lienarly independent for each $p\in M$, then $M$ is orientable.
The conversely of the above result is not true if $n-m>1$. So I want to find some example of an orientable surface that does not admit such kind of vector fields, I searched about that but it was fruitless. Someone know where can I find such a example?