You said: "I'm explicitly looking for a Classless, skill-based system..."

Years ago, decades actually, White Wolf inc. came out with the Storyteller system in the 1990s. It is mostly a classless, skill-based system. You will still need to homebrew your own mechanics where the Storyteller system doesn't fit. White Wolf inc. then streamlined the rules in 2004 and renamed it the Storytelling System.

Don't know how you feel about generic game systems, but they tend to be skill-based and classless, like GURPS, BRP, D6 System, and FUDGE#Game_mechanics). No matter which way you go with this info, I believe you will need to homebrew some mechanics to round off the edges.

PS: All links lead to Wikipedia pages.