In one of my assignments, I was asked to prove that $\mathbf{u}$, $\mathbf{v}$, and $\mathbf{w}$ are in $\operatorname{span} (\mathbf{u},\mathbf{v}, \mathbf{w})$, so I want to define a set in set-builder notation for the spanning set as the sum of $\alpha\mathbf{u}$, $\beta\mathbf{v}$, and $\gamma\mathbf{w}$ for all scalars $\alpha$, $\beta$, and $\gamma$. The only problem is it doesn't define a universal set/space (like it doesn't say anything about them being in $\mathbb{R}^3$), so I can't use $\alpha \in \mathbb{R}, \beta \in \mathbb{R}, \gamma \in \mathbb{R} $ as the predicate. Can I use the universal quantifier in the predicate like this, or should it be written differently?
$$ \{ \alpha\mathbf{u} + \beta\mathbf{v} + \gamma\mathbf{w} : \forall{\alpha, \beta, \gamma} \} $$