"I was very proud to have played more than 50 times for my country and to have worn the armband, so to manage England will be a huge responsibility and a huge honour for me," Southgate told The FA's website.
"It's a difficult circumstance for me, because we've lost two senior coaches in the last few months who have both been very good to me, so that complicates my feelings. I certainly wish them both all the best for the future.
"With my experience with the Under-21s in international football, and having worked with so many of the players who have moved up to the seniors as well, I think we are in a good position to prepare for these games."
Speaking about the decision to dismiss Allardyce, Southgate added: "It's obviously been a difficult situation for The FA but it was important that there was some stability and continuity for everybody.
"So, from my point of view, it was important to step forward and give us the best possible chance to win these games. The fact that I know so many of the support team and the players already is really important, given the time that we have to prepare for these matches."
Southgate's first game in charge will be the World Cup qualifier against Malta at Wembley on October 8.