I know how to build the Haar measure somewhat explicitely on Lie groups (via differential forms) and profinite groups (by using the lemma that open subsets of a profinite group are unions of cosets of open sets + the observation that an open subgroup has measure the inverse of its index, or as an inverse limit of the counting measures). (I think, vaguely, that the same works for locally profinite groups.)
Are there other classes of locally compact Hausdorff groups where the Haar measure can be built in an "explicit" way (not through abstract nonsense)? Are there examples of locally compact Hausdorff groups where the general existence argument is more useful than a direct construction for that case?