I want to show that the application
$$\phi: \mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R} \quad \text{given by} \quad \phi(u, v)=u \times v$$
It is bilinear
In first place, note that
\begin{align*} \phi(\alpha u+ \beta v), w &= (\alpha u+ \beta v) \times w \\ &=\begin{vmatrix} i & j & k\\ \alpha u_{1}+\beta v_{1} & \alpha u_{2}+\beta v_{2} &\alpha u_{3}+\beta v_{3} \\ w_{1} & w_{2} & w_{3} \end{vmatrix} \\ &= \alpha \begin{vmatrix} i & j & k\\ u_{1} & u_{2} & u_{3}\\ w_{1} & w_{2} & w_{3} \end{vmatrix}+\beta \begin{vmatrix} i & j & k\\ v_{1} & v_{2} &v_{3} \\ w_{1} & w_{2} & w_{3} \end{vmatrix} \\ &= \alpha(u \times v)+ \beta(v \times w) \end{align*}
And you have one of the conditions to be bilinear, my question is with the other condition, it seems that if I verify that the application is symmetrical it would be quickly, but I do not see at all how to determine if it is symmetrical, any suggestions?