A Dash 8 departed with the fueling door open and the fuel cap off. A passenger advised of the problem but only after in flight. Some damage occurred and the fuel cap was lost.

Narrative

Fueler left fuel cap and fuel door open but turned the fueling to the off position. The off position gives us no indication in the flight deck that the fuel door or cap is in an unsafe condition. My First Officer informed me that the pre-flight inspection was completed and that we were ready to board. Normal boarding and takeoff procedures followed. In flight; Flight Attendant informed us that a passenger noticed the fuel door and fuel cap were open and the fuel cap was banging into the nacelle. No action was taken since there was no emergency evident and the flight would continue as normal to its scheduled destination and closest airport. We arrived at our destination and during re-fueling and post-flight walk around; it was noticed that the fuel cap was missing. Maintenance arrived and noticed the damage that the fuel cap had caused and they grounded the airplane to be fixed.The First Officer was mistaken and never did a proper pre-flight even though I was informed otherwise. Since I was told that the pre-flight was complete; no further action was needed. However; the fueler left the fuel door and cap off which is against their procedures. I was given the go ahead as if everything was normal and completed but later discovered that the ground crew did not accomplish their pre-flight walk around before I started the right engine. The way to prevent this from happening is for everyone to perform their task assigned and report accurately to the Captain. I was told by the First Officer and the ground personnel to continue the flight as normal as if all procedures were completed properly.