Is this a group?

Algebra Level 3

For vectors (x1,y1,z1)(x_{1}, y_{1}, z_{1}) and (x2,y2,z2)(x_{2}, y_{2}, z_{2}) in R3\mathbb{R}^{3}, define the cross product ×\times as such:

(x1,y1,z1)×(x2,y2,z2)=(y1z2z1y2,z1x2x1z2,x1y2y1x2)(x_{1}, y_{1}, z_{1}) \times (x_{2}, y_{2}, z_{2}) = (y_{1}z_{2}-z_{1}y_{2}, z_{1}x_{2}-x_{1}z_{2}, x_{1}y_{2}-y_{1}x_{2})

Does R3\mathbb{R}^{3} form a group under the cross product?

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