We propose a new class of game-theoretic models for network formation in which strategies are not directly related to edge choices, but instead correspond more generally to the exertion of social effort. The observed social network is thus a byproduct of an expressive strategic interaction, which can more naturally explain the emergence of complex social structures. Within this framework, we present a natural network formation game in which agent utilities are locally defined and that, despite its simplicity, produces a rich class of equilibria that exhibit structural properties commonly observed in social networks - such as triadic closure - that have proved elusive in most existing models. Specifically, we consider a game in which players organize networking events at a cost that grows with the number of attendees. An event's cost is assumed by the organizer but the benefit accrues equally to all attendees: a link is formed between any two players who see each other at more than a certain number r of events per time period. The graph of connections so obtained is the social network of the model. We analyze the Nash equilibria of this game when each player derives a benefit a>0 from all her neighbors in the network and when the costs are linear, i.e., when the cost of an event with L invitees is b+cL, with b>0 and c>0. For a/cr > 1 and b sufficiently small, all Nash equilibria have the complete graph as their social network; for a/cr < 1 the Nash equilibria correspond to a rich class of social networks, all of which have substantial clustering in the sense that the clustering coefficient is bounded below by the inverse of the average degree. Additionally, for any degree sequence with finite mean, and not too many vertices of degree one or two, we can construct a Nash equilibrium producing a social network with the given degree sequence.