Kramer, who won 5,000m gold last week, was the hot favourite and he lived up to his billing by powering around the Richmond Oval more than four seconds faster than Lee in the gruelling race.
He thought he had won but a lane infringement with eight laps left saw him disqualified, ending his dream of completing the first golden treble by an Olympic speedskater since Norway's Johann Olav Koss in 1994.
Kramer, the 23-year-old son of two-time Dutch Olympic speedskater Yep Kramer, threw his goggles down in disgust on Tuesday and was a forlorn figure as Lee celebrated an unexpected victory.
Lee took the title in 12:58.55, a new Olympic record, 3.52sec second clear of Skobrev and was ecstatic.
"I'm really happy that I'm one of the first Asian skaters to win this race," said the Korean.
"I don't think I will be able to sleep now that I have the gold medal."
Asked about the Kramer controversy, he said: "I want to compete with Sven again."
The victory meany Korea became the most successful speed skating nation in Vancouver, after Mo Tea-Bum won the Men's 500m title and Lee Sang-Hwa captured the women's 500m.
Kramer blamed his coach Gerard Kemkers for giving him the wrong information.
"Usually, I don't want to blame anyone else, but this time I can't do anything else. I wanted to go on the outer lane then just before the cone Gerard shouted 'inner lane,' I thought he's probably right and went to the inner lane," he said.
Kramer's Dutch teammate De Jong said it was a sad end, but he was happy to pick up a medal.
"It happens in speed skating," he said of Kramer. "It is really hard to skate that fast and he missed the crossover. It really sucks."
Skobrev paid tribute to the Dutch star, saying he deserved to win.
"Sven in my eyes is almost a hero. He's young, he's got talent and he's a hard worker. I tried to learn and follow him during this event," he said.
"It's not only about racing, or how you are in shape, your muscles, your feeling, it's also about your head, clear thinking and analysing every lap. Everybody sort of knew that he should come first and he certainly deserves the gold."