As OT pointed out, the 2 tasks require 2 different blade types generally. CAN you perform both tasks with a single blade type? Yes you most certainly can. But when it comes to larger game animals such as Elk and Moose, as you found out last season, you should have the proper tools for the task. A good curved edge "skinning" knife is required for Elk and larger game. This one from Gerber is highly recommended.
http://www.gerbergear.com/Activity/H...nife_31-002743 It's form lends to making the skinning MUCH easier. For quartering I recommend a fixed strong back blade of 4 inches or more such as the Gerber Gator seen here
http://www.gerbergear.com/Knives/Fix...de-Knife_46904
And before you ask, no I am not affiliated with Gerber in any way. They just make pretty dang good knives that tend to take and hold an edge (when sharpened properly) as well, if not better than, most knives on the market today. When I pack for an Elk or Moose hunt, I generally have 4 knives with me. I tend to bone out the meat at my camp site so I use a filet knife as well. Saves on weight and packing space. But for the job of skinning and quartering you can get away with 2.