If ker ${\cal M}^{T} $ is one-dimensional, there are two possibilities:
Case 1: $(\mathbf{u},\mathbf{v},M\mathbf{u})$ are linearly dependent; then $M\mathbf{v} $ is linearly independent of the other three, and we can construct the vector $\mathbf{w} $ orthogonal to the others via $\mathbf{w}_{i} = \epsilon_{ijkl} \mathbf{u}_{j} \mathbf{v}_{k} (M\mathbf{v} )_{l} $.
Case 2: $(\mathbf{u},\mathbf{v},M\mathbf{u})$ are linearly independent; then $M\mathbf{v} $ is linearly dependent on the other three, and we can construct the vector $\mathbf{w} $ orthogonal to the others via $\mathbf{w}_{i} = \epsilon_{ijkl} \mathbf{u}_{j} \mathbf{v}_{k} (M\mathbf{u} )_{l} $.
We can furthermore consolidate the two cases into the expression $$ \mathbf{w}_{i} = \epsilon_{ijkl} \mathbf{u}_{j} \mathbf{v}_{k} [(M\mathbf{u} )_{l} + (M\mathbf{v} )_{l} ] $$ because in either case, it reduces to the expression specific to that case already given.