Suppose $\mathbf{v}_1$ and $\mathbf{v}_2$ are real vectors of length $N>3$.
If we use Gram-Schmidt process, we can find two orthogonal vectors $\mathbf{u}_1, \mathbf{u}_2$ such that $$ \mathbf{u}_1 = \mathbf{v}_1\ , $$ $$ \mathbf{u}_2 = \mathbf{v}_2 - \frac{<\mathbf{v}_2,\mathbf{u}_1>}{<\mathbf{u}_1,\mathbf{u}_1>}\mathbf{u}_1\ , $$ where $<\mathbf{x},\mathbf{y}>$ denotes the inner product between the two vectors $\mathbf{x}, \mathbf{y}$.
But I am wondering whether it is possible to find a vector $\mathbf{z}$ that is mutually orthogonal to $\mathbf{v}_1, \mathbf{v}_2$.
If we are dealing with vectors $\in \mathbb R^3$ we can use cross product to obtain
$$z=v_1 \times v_2$$
otherwise we can proceed again by G-S selecting any $v_3$ which is not in the span of $v_1$ and $v_2$.