Haar measure from a left-invariant coframe ($d\mu =|{\theta }^1\wedge \cdots \wedge {\theta }^d|$)

On a $d$-dimensional Lie group, take the coframe dual to a basis of left-invariant vector fields. Its top wedge is nonvanishing and invariant under left translations, so its absolute density defines a left Haar measure. If the frame has coordinate matrix $A(x)$, the density is $|\det A(x){|}^{-1}d{x}^1\cdots d{x}^d$. This makes explicit computation of Haar measure a determinant calculation.