Product of $2n$ factors with omega

If $1$, $\omega$, ${\omega }^2$ are the three cube roots of unity, prove that $(1-\omega +{\omega }^2)(1-{\omega }^2+{\omega }^4)(1-{\omega }^4+{\omega }^8)\dots \text{to }2n\text{ factors}=2^{2n}$.

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. VIII. Surds and Imaginary Quantities, Examples VIII.b, No. 28