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. 

EDIT. Reading a little farther into the Gardner essay, I see he writes, 

The only published proof known to me that the conjecture is true is given by G. J. Chang, M. C. Hu, K. W. Lih and T. C. Shieh in "Exact Difference Triangles," Bulletin of the Institute of Mathematics, Academia Sinica, Taipei, Taiwan (vol. 5, June 1977, pages
191- 197).

This paper can be found at http://www.math.sinica.edu.tw/bulletin/bulletin_old/51/5120.pdf and the review is MR0491218 (58 #10483).