To clarify --- you mean the graph on n vertices having n(n-1) directed edges?  But what do you mean by a 1-factor?  A set of edges that gives (indegree+outdegree)=1 at each vertex?  (that is, a set of edges which when undirected forms a perfect matching?

Or do you mean a set of cycles which meets each vertex exactly once?

ัThe standard notion of matching applies only for undirected graphs. Given a directed graph, you can converse the graph to the undirected one. An edge {u,v} in the undirected graph appears if and only if both (u,v) and (v,u) are edges in the original directed graph.

Enumerating matchings in the complete undirected graph is easy: and in fact, in the directed version, almost easier.  Enumerating covers of the complete directed graph by directed cycles is not hard either --- though the answer is not as simple as for matchings.  It becomes a sum over stirling numbers instead of just essentially a binomial coefficient.

This can be solved for any directed graph, not just complete directed graphs.We convert it to a max flow problem, ala Ford and Fulkerson algorithm.  Convert each vertex into two vertices.  V becomes Vin  and Vout.  now each directed arc VW is entered as the arc Vin Wout.  Add two more vertices, a source that has arcs to all the "in"  vertices and a sink receiving arcs from all "out"  vertices.  Give all arcs a capacity of 1, Run the max flow min cut algorithm to find the max flow from source to sink.  The max flow incorporates a max matching, which may or may not be a perfect matching whne none exists.  This algorithm is order n squared.