In weighted congestion games, players’ weights and resource functions are predefined. This way, they can be applied for modeling traffic intensity, exploring market competition, and analyzing other problems with a congestion effect. In some normal-form games however, players’ weights and resource functions are not defined explicitly and players may be unaware of their existence. This article finds the necessary and sufficient conditions for representing a normal-form game as a weighted congestion game. Axioms are formulated that guarantee there exist positive weights of players and positive-definite resource functions. It is proved that a normal-form game satisfies the axioms of Positivity and the Independence of Irrelevant Choices (Konishi et al., 1997) if and only if it is a singleton weighted congestion game. This result indicates that the payoff functions of players in hedonic games are represented in the form of a weighted congestion game. It is demonstrated that a normal-form game satisfies the axioms of Non-Negativity, Transfer, Resource Marginal Contribution, and the Independence of Irrelevant Choices if and only if it is a full weighted congestion game with player-independent resource functions.