Solvability of matrix filling
This is a partial solution.  Such a matrix exists if n is a prime power.

Let F be the finite field of order n.  We construct an n×n(n−1) matrix whose rows are labeled by F, whose columns are labeled by (F∖{0})×F, and whose entries are in F as follows: the i-th row of the column labeled (a, b) is given by ai+b.  In words, each column corresponds to a degree-one polynomial in F.  Then each column contains each element of F exactly once, and no two columns have equal entries in more than one row because the values of two distinct degree-one polynomials can coincide at at most one point.

(If you want a matrix whose entries are in {1,…,n} instead of in F, replace the elements of F with {1,…,n} arbitrarily.)