Equivalence of Definitions of Vector Cross Product

$(1)$ implies $(2)$
Let $\mathbf a \times \mathbf b$ be a vector cross product by definition $1$.

Then by definition:
 * $\mathbf a \times \mathbf b = \paren {a_j b_k - a_k b_j} \mathbf i - \paren {a_i b_k - a_k b_i} \mathbf j + \paren {a_i b_j - a_j b_i} \mathbf k$

Thus $\mathbf a \times \mathbf b$ is a vector cross product by definition $2$.

$(2)$ implies $(1)$
Let $\mathbf a \times \mathbf b$ be a vector cross product by definition $2$.

Then by definition:
 * $\mathbf a \times \mathbf b = \norm {\mathbf a} \norm {\mathbf b} \sin \theta \, \mathbf {\hat n}$

Thus $\mathbf a \times \mathbf b$ is a vector cross product by definition $1$.