Why does the multiplication in a division algebra depends on every component?

Question

In a division algebra A over $\mathbb{R}$ we have this multiplication (A isomorphic to $\mathbb{R}^{n}$) $$\mathbb{R}^{n} \times \mathbb{R}^{n} \to \mathbb{R}^{n}:(x,y)\mapsto y=x\cdot y$$ where every component of z depends bilinear on every component of x and y. Why is it impossible that z just depends on every odd component of x or something like that? Is it because otherwise we get a problem with a zero divisor?