Yes, you can use flame blade in Wild Shape...
You can definitely cast the spell and then continue concentrating on it while in wild shape form. Furthermore, you are allowed to take any actions granted to you by the spell (specifically, evoking the blade and attacking with it) while you concentrate on it, thanks to this feature of Wild Shape (emphasis mine):
You can’t cast spells, and your ability to speak or take any action that requires hands is limited to the capabilities of your beast form. Transforming doesn’t break your concentration on a spell you’ve already cast, however, or prevent you from taking actions that are part of a spell, such as call lightning, that you’ve already cast.
So just like you can continue using your action to call down lightning bolts while concentrating on call lightning in your wild shape, you can use your action to attack with your flame blade while in wild shape.
... But only if your wild shape can hold a blade
However, the part where you run into problems is
If you let go of the blade, it disappears
This means that if you wild shape into a wolf, the blade is going to disappear, and even if you evoke it again with a bonus action, it's going to disappear again, because you have no way to hold it in your paws. However, if you were to wild shape into an ape, which has hands with opposable thumbs, I see no reason that you wouldn't be able to wield the blade as normal. Since you retain your wisdom score, you would even use the same attack modifier when attacking with it, since it is a melee spell attack (and not a melee weapon attack).
If you want to turn into a giant owl with flyby and make strafing runs wielding the flame blade in your talons, I'd recommend asking your DM first.