We know the following results:
Theorem 1: If a suface $M^m \subset \mathbb{R}^{m+n}$ of co-dimension $n$ admits $n$ linearly independent continuous normal vector fields $v_1, \dots, v_n:M \rightarrow \mathbb{R}^{m+n}$ then $M$ is orientable.
Also, the converse of the above result is true if the co-dimension of $M$ is one, more precisely:
Theorem 2: Every orientable surface $M$ of co-dimension $1$ in $\mathbb{R}^{n+1}$ admits a normal continuous non-vanishing vector field.
My question: I would like to know a example of a orientable surface $M \subset \mathbb{R}^{m+n}$ of co-dimension $n>1$ which has no vector fields $v_1, \dots, v_n$ satisfying the properties of the first Theorem 1.