A concrete model for Ω^∞ applied to Thom spectra (which is all what we need because Thom spectra are connective)
was given by Quinn in his thesis.

Very roughly, Ω^∞MG is a simplicial set whose n-simplices are n-dimensional G-manifolds with corners, with corners of codimension i assigned to simplicial faces of Δ^n of codimension i.

For precise definitions and statements see Quinn's paper
[“Assembly maps in bordism-type theories”](http://www.maths.ed.ac.uk/~v1ranick/papers/quinnass.pdf),
especially Sections 3 and 6.

Laures and McClure explain how to promote these simplicial sets
to strictly commutative symmetric ring spectra.
See their papers [“Multiplicative properties of Quinn spectra”](https://arxiv.org/abs/0907.2367)
and [“Commutativity properties of Quinn spectra”](https://arxiv.org/abs/1304.4759).