To understand skilligimink, one must read the original sources. Here’s a prime example from the book Sammie and Susie Littletail, by Howard R. Garis, chapter XXII – Sammie Colored Sky-Blue-Pink. It’s a little long but I think it tells what needs to be said.

Mamma Littletail prepares to dye Easter Eggs:

Well, you should have seen all the colors she had! There was red and blue and yellow and green and purple and pink and old rose and crushed strawberry and ashes of roses and magenta and Alice blue and Johnnie red and Froggie green and toadstool brown and skilligimink. That last, the storekeeper told Sammie, was a new color, very scarce. As there isn’t any more of it at the store, I can’t just tell you what it looked like, except that it was a very fine color indeed, Oh, yes!

Well, Sammie and Susie helped their mamma dip the eggs in the dye and stained them all sorts of pretty colors. Some were all one shade, and some were half one tint and half another, and then there were some all speckled with different colors, and very hard to make. Then, after they were all dry, Nurse Jane Fuzzy-Wuzzy, with her sharp teeth, just like chisels that a carpenter uses, drew pretty things on the eggs; pictures of trees and birds and mountains and flowers and fairy castles and lakes and hills, and all sorts of things. Oh, they were the prettiest Easter eggs you ever saw!

“Here is the last egg,” said Sammie. “May I dip this one in, mamma?”

“Yes,” she answered, but she never would have let him if she had known what was going to happen.

“I’ll make this a skilligimink color,” said Sammie, and he stood over the pot. Then, what do you think occurred? Why, Sammie leaned too far over and he fell right in that pot of skilligimink color; he and the egg together. And oh, dear me! what a time there was. He splashed around and scattered the skilligimink color all over the kitchen, and when his mamma and Susie fished him out, if he wasn’t dyed the most beautiful sky-blue-pink you ever saw! Oh, but he was a sight! The skilligimink color made him look like a piece of the rainbow. “Oh, Sammie!” cried Susie, “how funny you do look?” And Sammie grunted: “Huh! I guess it’s nothing to laugh at!” So they dried him with a towel, but the color didn’t come off for ever so long, honest it didn’t. But they had a lovely lot of Easter eggs, anyhow, ready for the children, and so Sammie didn’t mind much.

From Project Gutenberg‘s Sammie and Susie Littletail by Howard Roger Garis. (emphasis added)