# maximum number of perfect matchings of $K_{2n}$ such that no edge appears in two different matchings

Given a complete graph $$G=K_{2n}$$, construct a set $$P$$, such that each element $$p$$ of $$P$$ is a perfect matching of $$G$$ and every two elements $$p_i,p_j$$ don't share a common edge of $$G$$.What's the maximum cardinality of $$P$$?

I'd like to calculate the maximum number of perfect matchings of a complete graph of order $$2n$$, such that no edge appears in two different matchings and find an algorithm to construct one enumeration of such perfect matchings if possible.

I believe each such set could contain up to $${{2n}\choose{2} }/n = 2n-1$$ perfect matchings, but I was unable to construct one or prove such enumeration must exist.