(CNN) Having been so close to clinching her maiden grand slam title for so many years, but ultimately always falling short, many thought the increasing pressure of Caroline Wozniacki's near misses would see her never win a major.

The Dane went some way to answering those critics with a victory at the year-end WTA Finals in October, before sweeping all before her to win the Australian Open in January.

With that first grand slam now under her belt, Wozniacki herself admits the alleviation of expectation could potentially now propel her to become a multiple major winner.

"Going into the next three grand slams, I definitely have more belief and I think less pressure on my shoulders," Wozniacki told CNN Sport's Alex Thomas.

READ: Caroline Wozniacki beats Simona Halep to win Australian Open and first major

"That can be a great combination and I'm just trying to work hard every day to peak for the French (Open) and obviously Wimbledon and the US Open. I'll definitely do my best and we'll see what happens."

Despite the French Open and Wimbledon being just around the corner, Wozniacki has understandably already set her sights on the year's final grand slam, the US Open.

The 27-year-old has managed a combined total of just two quarterfinals and a handful of fourth round appearances at Roland Garros and Wimbledon since 2007.

Her fortunes have significantly improved at the US Open, where she's reached the semifinals three times and the final twice.

"I definitely feel like that's the one I was the closest at and I feel those courts really fit me and suit me well," she said. "I have great fans there and that was the one I thought I was going to catch first.

"But we'll see, hopefully I have a chance this year and hopefully if the draw provides good luck for me then I can go all the way."

READ: Caroline Wozniacki beats Venus Williams for first time to win WTA Finals

'Margarita on the beach'

Wozniacki is good friends with Serena Williams, who at the age of 36, after overcoming pregnancy-related health complications, could make her grand slam return at the French Open.

Already with an open era record of 23 majors to her name, former world No. 1 Williams has dominated women's tennis for almost two decades.

Though this longevity is something Wozniacki admires, arguing it is what makes Williams one of sport's all-time greats, she herself cannot imagine playing professional tennis for so long.

"Definitely not, I don't see myself playing for that many years," she said.

"I think it's incredible that she has been playing (for so long) and playing so well for so many years and obviously that's why she's one of the greatest women's athletes ever and has made so much history.

"I see myself by that point probably just enjoying my time with a margarita on the beach."

READ: Serena return sparks seeding debate

'I definitely didn't sleep well'

But 2018 year hasn't been all positive for Wozniacki. Just last month, while playing at the Miami Open, she claimed members of her box, which included two children, had been subject to death threats from the crowd.

Facing Puerto Rican Monica Puig, who was backed by partizan home crowd -- Miami has a large Latin population -- Wozniacki said some of the support crossed the line.

She criticized the organizers of the Miami Open in a post on Facebook and found the response by the tournament director, former player James Blake, less than satisfactory.

At the time, tournament director Blake issued a statement, which said: "We had tournament and Women's Tennis Association staff as well as tournament security courtside. They never witnessed, nor were they notified of any specific threats made to their players or their families."

It's a response that still rankles with Wozniacki.

"I can't really do anything about how they reacted to it," she said. "Obviously I don't think they did a great job so I was a bit disappointed with their statement and obviously they didn't do anything about it.

"But I was proud of the way I stood up for myself and my family and having my fiance's 10-year-old niece and nephew there, crying after the match and having to explain to them on the car ride home that this is not how people normally behave was very hard for me.

"I definitely didn't sleep well for a while after that. I think you live and you learn and the beauty of tennis is that we normally have incredible fans all over the world and they're very respectful usually and I hope this is just a one-off thing and that it won't happen again."

Blake's representatives did not immediately respond to CNN's request for comment.