I'm a huge fundamentals and basics person-I was very much this way throughout my time in the Army too, so don't feel like I am picking on you if I say that there's one thing or the other to do differently. The push up, from what I've seen, is one of the two fundamental movements (the squat being the other) where I see there is a tendency for a breakdown in motor pattern and mechanics.
I will draw on Kelly Starrett's (K Star) mobility WOD video from last January. He hosts Dan Bailey and Rich Froning, who are absolute physical beasts and have spent immense amounts of time maximizing physical outputs and mechanical efficiency.
"In this video example, you will notice that both Rich and Dan have a vertical forearm during both their pushup and in the bottom of their burpee. This vertical forearm is the same they would use if you flipped them over and had them bench or even perform a hand stand push up. The vertical forearm is just coaching cue that really means full mechanical advantage of the triceps and pecs as well as correct load sequencing at the shoulder (easier to cue on the vertical forearm)." - K Star
So check this out, we talk about vertical shins in our squats and here, we talk about about vertical forearms in the push up- coincidence? Not really...we develop full mechanical advantage in these positions. You can see that you are capable of executing a push up with your hands externally rotated to the point where your fingers are facing to your rear-check out the video. Personally, I am incapable of this movement currently because my wrists and forearms are extremely tight. Yes, yet another reason to be on top of mobilizing. :)
That aside, don't be upset if i tell you to: get external rotation on your shoulders when you set up your push up and keep your elbows close such that your thumbs are by your nipple-line at the bottom position. Yes, the elbows out during your push up can be a more comfortable or familiar position- hell, all of the push ups I did for my physical fitness test while I was in the Army were ALLLLLLLLL elbows out. Was it right? Nope. Did I suffer for it? Yup! I sure effing do. Old habits die hard, so I have to be deliberate with my mechanics with this; it's not so hard if you don't start with bad habits.
Easier does not necessarily mean better.
A. 20 mins to max Front squat
B. 3x5 FS @ 80% above
7 min AMRAP
20 KBS 55/35
10 Push ups