I have noticed something which may have been noticed already: take a match, and light it, then hold it so that the flame is directly above your hand. The flame will eventually go out. My guess as to why this happens is that the convection causing the flame to go upward eventually starves the flame of oxygen, so it gets extinguished, but I don't know for sure.
Although it looks like the wood of the matchstick is burning, this isn't the case. Combustion is a gas phase reaction, and what actually happens is that the heat of the flame heats the wood and decomposes it so it gives off various flammable gases. These then react with oxygen to give the flame.
So sustaining a flame requires that the burning material be heated enough to decompose or vapourise. For example the reason a match goes out when you blow on it is because your breath cools the wood and prevents it decomposing.
When you hold a match upwards the flame travels up away from the unburnt wood, so the wood is not heated as much as with the match held horizontally. Depending on exactly how the match is designed the heating of the unburnt wood may be insufficient and the match will go out. You'll find a sufficiently thick match will continue to burn because it does generate enough heat, while a thin match with a correspondingly small flame will go out.