Define a sequence of non-constant random variables on the probability space $([0,1],\mathscr{F},P)$ that are pairwise independent?

Question

It's easy to construct two independent random variables on $[0,1]^{2}$. For example, $f(x,y)=x,g(x,y)=y$.

How to define a sequence of non-constant random variables $\{f_{n}\}_{n=1}^{\infty}$ on the probability space $([0,1],\mathscr{F},P)$ that are pairwise independent?