This is the first problem in Chapter 9 of Martin Gardner, Penrose Tiles to Trapdoor Ciphers. In the addendum to the chapter, he writes that Herbert Taylor has proved it can't be done for $n\gt5$. Unfortunately, he gives no reference.
There may be something about the problem in Solomon W Golomb and Herbert Taylor, Cyclic projective planes, perfect circular rulers, and good spanning rulers, in Sequences and their applications (Bergen, 2001), 166–181, Discrete Math. Theor. Comput. Sci. (Lond.), Springer, London, 2002, MR1916130 (2003f:51016).
See also http://www.research.ibm.com/people/s/shearer/dts.html and the literature on difference matrices and difference triangles.