Vatanen played the past three seasons in New Jersey after being traded by the Anaheim Ducks on Nov. 30, 2017 for forwards Adam Henrique, Joseph Blandisi, and a third-round pick in the 2018 NHL Draft.
Selected by Anaheim in the fourth round (No. 106) of the 2009 NHL Draft, Vatanen has 194 points (45 goals, 149 assists) in 434 regular-season games and has 23 points (six goals, 17 assists) in 44 Stanley Cup Playoff games.
Skjei is in the second season of a six-year contract. The 25-year-old has 23 points (eight goals, 15 assists) in 60 games this season.
"Skjei is a big body (6-foot-3, 210 pounds), good defender," Waddell said. "He's going to add a little offensively. He plays the game hard, the right way."
A first-round pick (No. 28) in the 2012 NHL Draft, Skjei has 112 points (25 goals, 87 assists) in 307 regular-season games and seven points (four goals, three assists) in 17 playoff games.
"I will say, trading Brady Skjei is difficult," New York general manager Jeff Gorton said. "He's a special guy. He's a really good person. He's a good player. He's been a good Ranger. We wish him nothing but the best. It's just a move that we felt we had to make."