I know about the notion of orientation in an abstract vector space, yet I am stuck on this seemingly trivial issue.
Let $V$ be a subspace of $\mathbb{R}^{d >3}$. Suppose that $\dim V = 3$. If I have two orthonormal vectors $e_{1}, e_{2}$ in $V$, then there are two canonical ways to define an orthonormal basis of $V$: pick $e_{3}$ so that $(e_{1}, e_{2}, e_{3})$ follows – or does not follow – the right hand rule.
Note that I can define $e_{3}$ without any reference to $\mathbb{R}^{d}$. In other words, if I have to repeat the process of completing a pair of vectors into a basis for different three-dimensional subspaces, then I can perform such task consistently: I can choose to follow (or not follow) the right hand rule once and for all.
On the other hand, when $\dim V = 2$ the situation is different. If I have multiple two-dimensional vector subspaces, then I have no means to compare the choices of orientation that I make on each of them.
Could anybody explain why this happens? Is it simply because the cross product in dimension two does not exist?
Based on clarifications in the comments: If $V$ and $V'$ are complementary finite-dimensional subspaces of a real vector space, then orientations on any two of $V$, $V'$, and the direct sum $V \oplus V'$ determine an orientation on the third.
An orientation on any one space, by contrast, does not determine an orientation on either of the other two.
Perhaps surprisingly, "the right-hand rule" does not "promote" an orientation on a plane $P$ to an orientation of a containing three-dimensional space $V$: The meaning of "right hand" amounts to an orientation on the three-space $V$. Martin Gardner's The New Ambidextrous Universe (pp. 151 ff., especially 153-154 of the 1990 edition), discusses the issue in detail, including Immanuel Kant's (temporary) belief that a (chiral) right hand could only fit one arm of a (bilaterally symmetric) torso.