Since each row is an arrangement of n distinct objects / numbers, each row is a permutation and could be mapped to an element of a permutation group S_n. Those who are interested to study the details about the Permutation Group are referred to the Wikipedia Page here. If we consider an acyclic permutation a, then {1, a, a², ..., a^n-1} form a subgroup of S_n and they form the rows of an n by n Latin Square. Clearly no row has any duplicate since they are arrangements of distinct numbers. No column has a duplicate either. As otherwise the starting permutation a fails to be acyclic.

Clearly these do not include all the Latin Squares of Order n. In fact rows of an order n Latin Square need not even be a group. Still we may relax the condition of the starting permutation a to be acyclic and may consider to be product of r cycles each of length s where n = rs. Then also it is possible to work out n permutations that form a subgroup of S_n and eventually form the rows of a Latin Square of order n.

So if we start we a random permutation of n numbers, create the corresponding natural acyclic permutation out of it, then we get a random Latin Square of order n by considering its powers. Till now I don't know any other method of generating random Latin Squares more efficient than this.

Extendibility of such Latin Squares to its Orthogonal Mate is a different question.