Many variants, and formerly different methods of playing the game.
According to Randolph, Hofer, 1907, "Popular Folk Games," was the earliest he knew in print with "Skip-to-ma-Lou, My Children Dear."
There are a number of variants in Jour. American Folk-Lore.
This one is from Randolph, coll. 1927.

Lyr. Add: SKIP TO MY LOU

"After choosing partners, all the players join hands and form a large circle, while everybody sings":

Flies in the buttermilk, two by two,
Flies in the buttermilk, shoo fly shoo
Flies in the buttermilk, two by two,
Skip to my Lou, my darlin'.

"While this is being sung, one couple steps into the ring and chooses another boy, so that there are two men and one girl in the center, who hold hands and dance about the circle with a peculiar skip and double-shuffle step.
The first boy and girl hold their hands high, and the odd boy steps under the arch thus formed, after which the first couple joins the circle again. The boy left alone in the ring calls in another couple, then he and the girl make an arch and join the circle, leaving the new boy alone in the center. Then he chooses another couple, and so on."
Meanwhile, the song proceeds:

Little red wagon, painted blue, (3x)
Skip to my Lou, my darlin'.

Dad's old hat got tore in two, (3x)
Skip to my Lou, my darlin'.

Purty as a redbird, purtier too, (3x)
Skip to my Lou my darlin'.

Cain't get a redbird, a bluebird 'll do, (3x)
Skip to my Lou, my darlin'.

From Carl Durbin, Pineville, MO, 1927. With tune. No. 515, A, vol. 3.

Other verses from Arkansas and Missouri, also from Randolph:

She is gone, an' I'll go too,...
Git me another'n as purty as you, ...
Hair in the butter, six foot long, ...
Chicken in the breadpan, scratchin' out dough, ...
Rabbit in the briar parch, shooe, shoo, shoo, ...
Hole in the haystack, chicken fell through, ...
Hog in the cornfield, shoo, shoo, shoo, ...
Cain't git a fat gal, skinny gal'll do,...
Come along Maw, an' let's go visitin', ...
Cowboys a-leavin', two by two, ...
Bring in the biscuit, two by two, ...
One old boot an' one old show, ...
Choose your partners, skip to my Lou, ...
Cat's in the cream jar, what'll I do? ...
Right across center, two by two, ...
Stole my gal, that'll never do, ...
Skip, skip, why don't you skip? ...
Chicken in the dough tray, what'll I do? ...


Also see related game, "The Miller Boy," from England, versions in Gomme, "Traditional Games. Also found in America.
Tune different from 'Skip to My Lou." A tune for "Skip to My Lou," found in North Carolina, in Brown, North Carolina Folklore, Vol. 5, seems more akin to that of "The Miller Boy" than the tune general in the Ozarks and West.
From Arkansas:

Lyr. Add: THE MILLER BOY

Happy is the miller boy who lives by the mill,
The mill turns around with its own free will,
Hand on the hopper and the other on the sack,
Lady keeps a-going, gents turn back.

Snow and it blows and it's cold stormy weather,
Along comes a farmer a-selling apple cider,
You be the reaper and I'll be the binder,
Lost my true lover and here's where I find her.

Sailing east, we're sailing west,
We are sailing far o'er the ocean,
Any young man who wants a wife
Had better be taking a notion.

Coll. from Dr. G. E. Hastings, Arkansas, 1942.
"The Miller Boy," No. 518 variant E, Randolph, Ozark FolkSongs, vol. 3.

Randolph, in 1980, said, "The play-party is passing, and the next generation will probably see its total extinction in the Ozark country." The game has been preserved, in variant form, by square dance and other dance groups.

Verses in Austin E. and Alta S. Fife, 1969, "Cowboy and Western Songs," no. 99, p. 275, with score:

Lou, Lou, skip to my Lou, ...
Little red wagon, paint it blue, ...
She's gone again, skip to my Lou, ...
Fly's in the buttermilk, shoo, fly, shoo, ...
Cows in the cornfield, two by two, ...
I'll get another one better'n you, ...

Coll. from singing of Bill Koch, South Dakota and Kansas.

Lomax attribution of Lou to Scottish 'loo' is, of course, fakelore.