I took this photo in March 2013 when we visited our daughter who was attending college in Seoul. The entire trip I kept waiting to get a good view of that mountain. On our last morning, we went to Gyeonbok Palace. Finally got it--and with the bonus frame of that beautiful red gate!
This palace should be at the top of your list if you're visiting Seoul. While I loved the Changdeok Palace gardens, there's just a majestic feel about the size of Gyeonbok. It's right at the end of Sejongro, the main road through Seoul, and in front of the current residence of the president called The Blue House. (You can't get through behind the palace because of security for The Blue House, so be sure your cab driver knows where he's going.)
There are English language tours daily, as well as a colorful changing of the guard procession.