I have been playing Dragon Age: Origins and from what I(probably most off us I think) have heard about the story in Skyrim it is rather similar.
It breaks out a civil war and at the same time a dragon starts its invasion of the world(Alduin and Archdemon).
Am I just going crazy or have Bethesda taken some inspiration from DA:O?

well a civil war fits into a natural order of things once there is no king, and there was no king since oblivion which came out before DA:O

No civil war within OB though. Only the invasion seem to be the drive of the story.

doesnt matter if the civil war is in OB or not, civil war doesnt break out a week after a king is unexpectedly assassinated

tiber septim was the ruler of skyrim, with his death a civil war is something that youd expect . . . it happened in real life countless times, the only thing 'plagiarized' here is the history of humanity

I don't think they took inspiration, it's just a coincidence. The concepts for Skyrim's story may be older than Dragon Age.