Let \(V\) be a three-dimensional vector space over an algebraically closed field \(\mathbb{F}\) of characteristic different from \(2\). The goal of the paper is the classification of the non-degenerate alternating maps \(f:V{\times}V \to V\) over \(\mathbb{F}\) under the tensorial action of the full linear group \(GL(V)\) in \(\bigwedge^{2}V^{*}{\otimes}V\). The authors obtain this classification in terms of a scalar invariant \(\sigma_{2}(f)\) attached to each non-degenerate \(f \in \bigwedge^{2}V^{*}{\otimes}V\). The main result of the paper states that if \(f,f' :V{\times}V \to V\) are two bilinear alternating maps, then they are equivalent if and only if, either \(\sigma_{2}(f)=\sigma_{2}(f') \neq 0\), or \(\sigma_{2}(f)=\sigma_{2}(f') = 0\) and \(n_{f}=n_{f'}\), where \(n_{f}\) and \(n_{f'}\) are some discrete invariants attached to \(f\) and \(f'\).