Cain on "mission," and talks Defense sec'y. spot

One name that's missing from the Iowa caucuses is Herman Cain.

He suspended his presidential campaign last month in the face of escalating accusations of sexual misconduct.

He promised not to go away -- and he hasn't.

Video: Memorable moments from the campaign trail

Pictures: On the trail in Iowa

Cain said on "The Early Show" that, though he isn't in the running for president anymore, he hasn't "abandoned the mission" which, he said, "is to help get Barack Obama out of the White House."

He says he's starting a "movement," to "keep attention on solutions." He said, "One of the things that has frustrated me and many of the American people has been all of the negative attacks that have taken place during this campaign. ... Secondly, it takes attention away from how you're going to fix stuff. This is what the American people want. When I was in the campaign and I was at the top of the polls, it was the fact that people were connecting with solutions. And so what I plan to do is continue to promote solutions to problems, because not enough attention is being paid to them."

When asked if he endorsed any of the remaining candidates, Cain responded, "I'm not going to endorse anyone right now, but, when I do make an endorsement, it will be unconventional. Remember, I'm the unconventional candidate with unconventional ideas."

Cain has recently said if a Republican were to defeat Mr. Obama, he would be interested in the Secretary of Defense position in a new administration, CBS News' Chief White House correspondent Norah O'Donnell pointed out.

"That is a big job to serve as Secretary of Defense," she remarked to Cain. "I just want to ask you a couple questions. You had some mistakes on the campaign trail. You were confused about the Taliban. You said you thought the Taliban might take control in Libya. You know that the Taliban are not in Libya, correct?"

"Correct," Cain said, adding, "Norah, look, it's a learning process, and there are a lot of things that I have learned as a result of having been in the campaign, as well as after the campaign. Now, the reason that I indicated that I would consider being Secretary of Defense, I don't think that my qualifications of being the Secretary of Defense should be based on something I said months ago in error. No, it should be based upon leadership. National security is one of our biggest challenges that we have in this country, and I believe that, as a leader in that position, if we have the right president, I would be able to bring a lot to the party in terms of helping to prioritize those things that need to be prioritized."

CBS News chief Washington correspondent Bob Schieffer said, "Mr. Cain, wouldn't you have to start out being fairly familiar with the basic national security issues are? I mean, you throughout the campaign seemed unaware of a lot of the things that the Secretary of Defense would have to deal with. I mean, good intentions are one thing and bringing fresh insight. But do you really, sincerely, feel that you have the background and really the knowledge that you could handle a cabinet-level position like that?"

"Bob, allow me to remind you of something that didn't get picked up when I was beat up about what I didn't know. I served on the Citizens Advisory Board of the Strategic Air Command in Omaha, Nebraska, for several years when I was there. I was exposed to some of those issues at that particular time. I started my career as a ballistics analyst with the ... Navy. I've actually been on a Navy ship. So, I have been around some of these issues. So, it's not that I'm totally -- "

Schieffer interjected, "Sir, if I may interrupt. I have been on a Navy ship, too, but I don't think that qualifies me to be Secretary of Defense."

"Bob, I didn't say that qualified me to be Secretary of Defense," Cain said. "There you go...when I'm trying to make a point. That's not the point, Bob. I didn't say being on a ship one time qualifies me for being Secretary of Defense. My point is quite simply this: We need strong leadership in that position. And it was a hypothetical question and I gave a response based upon the question that was asked."

O'Donnell then posed this question: "How many people serve in the armed services?"