The analysis of a chemical reaction network by elementary flux modes is a very elegant method to deal with the stationary states of the system. Each steady state of the network can be represented as a convex combination of these modes. They are elements of the nullspace of the stoichiometry matrix due to the imposed steady-state condition. We propose an approach, which first derives the basis vectors of the nullspace and then calculates the elementary modes by an apt linear combination of the basis vectors. The algorithm exploits the special representation of the nullspace matrix in the space of flows and the fact that elementary modes consist of a minimal set of flows. These two ingredients lead to construction rules, which diminish the combinatorial possibilities to design elementary modes and, hence, reduce the computational costs. Further, we show that the algorithm also accounts for reversible reactions. If a system includes reversible reactions, it can be transformed into an unidirectional network by considering the forward and the backward flow separately. We derive a projection operator, which reveals the interrelationship between the two representations.