DGP where uniform random variables are pairwise independent but mutually dependent

Question

Consider $K=3$ random variables $X,Y,Z$. Could you suggest a data generating process such that

(1) $X,Y,Z$ are pairwise independent

(2) $X,Y,Z$ are mutually dependent

(3) $X\sim U([0,1])$, $Y\sim U([0,1])$, $Z\sim U([0,1])$

Could you generalise to any $K>2$?