To prove that $$\mathbf{v} \times \mathbf{u} = -\mathbf{u} \times \mathbf{v} $$ we can either use the right-hand rule.
If going from $\mathbf{v}$ to $\mathbf{u}$ is in a clockwise direction, then $\mathbf{v} \times \mathbf{u}$ points downwards. $-\mathbf{u}$ to $\mathbf{v}$ would also be in a clockwise direction, thus the result $-\mathbf{u} \times \mathbf{v} $ would also point downwards. The magnitude is same in both cases.
To prove that $$\mathbf{v} \times \mathbf{u} = -\mathbf{u} \times \mathbf{v} $$ we can either use the right-hand rule.
If going from $\mathbf{v}$ to $\mathbf{u}$ is in a clockwise direction, then $\mathbf{v} \times \mathbf{u}$ points downwards. $-\mathbf{u}$ to $\mathbf{v}$ would also be in a clockwise direction, thus the result $-\mathbf{u} \times \mathbf{v} $ would also point downwards. The magnitude is same in both cases.