Generalized construction of W basis

Although this question deals with the construction of a W-state, I was looking for a general way to find all the orthogonal W-states, given a number of qubits. For example, for three qubits, the first W-state I find is:

$$
W_3^1 = frac{1}{sqrt{3}}(|001rangle + |010rangle + |100rangle).
$$
An orthogonal state to this state would be (from this paper, page 4):
$$
W_3^2 = frac{1}{sqrt{3}}(|001rangle – |010rangle + |111rangle).
$$
But I am not sure whether this state qualifies as a W-state or not. I want to find all the other 6 orthogonal basis states like this. Also, I would like to be able to generate such orthogonal basis states in any dimension. Of course, I can use Gram–Schmidt process to find a set of orthogonal vectors. But I’m not sure whether they would be W-states or not. What is the proper way to generate such W-basis states given a number of qubits? TIA.