I want to know if sketching is a drawing skill, and if can I use drawing exercises to develop my sketching skill.
It depends on the definitions of some of your words.
If by sketching you mean the random doodling without goal that many people engage in as a tension release, then probably not. Yes, such doodling is practice at pencil control and may allow you to discover some interesting curve sequences and shading techniques, but it isn't going to help you develop your artist's eye. And since it is pursued without goal, the end product doesn't have to stand up to rendering criteria. If you start doodling with the image of a horse in your mind and later decide that is looks more like a dog, then you are no closer to gaining the ability to deliberately render horses.
If on the other hand your definition of sketching has goal, if you call it sketching (as opposed to drawing) because you intend to keep it locked up in a sketch book that you never show to anyone, then this kind of sketching will immensely assist you in learning to draw. Sketches aren't sacred, so when you make a mistake in a sketch, you can not only erase and fix it, you can write down on the blank space on the page exactly what you did wrong and how you corrected it. If the mistake is too big, you can move a few inches over on the page and leaving the earlier attempt in place to document the failure, start again. As long your sketching is an attempt to render a specific object or scene, then your sketching is adding to your ability to draw specific objects and scenes in a most valuable way.
All that having been said, it all depends on your definition of the words "drawing skill". If your intention is to learn how to make abstract but beautiful patterns on the page without concern for how well your creation reflects real life, if your goal is to make people feel something when they look at your creation even if they never truly figure out what it is you were trying to portray, if your definition of drawing has nothing to do with rendering reality, then everything that I have offered here is worthless and you will have to explore your art without my assistance to find your answer.
Sketching is a form of drawing. Often a drawing starts out as a sketch. So, yes, you can definitely do drawing exercises to develop your sketching skills - the more so because many drawing exercises consist of sketching something, like the basic shapes, or the construction of shadows or perspective lines.
But your question brings up two interesting points:
- It is hard to tell when a sketch becomes a drawing.
A sketch can be defined as an unfinished drawing, but has become a form of art in its own right, likely due to its expressionistic character.
- Sketching is usually not really regarded as a skill itself.
This probably has to do with the fact that while a sketch usually leads to drawing, it is a lot more free-form and idiosyncratic.
These matters are tied in with the fact that drawings, and especially sketches, only came to be regarded as meaningful, interesting, and - especially - valuable artworks themselves after during the renaissance the idea of the artist as an individual was introduced (through humanist ideology, think about the concepts of the genius and the homo universalis, for example). But for most of the history of the visual arts most drawings only formed the preparatory steps for paintings or sculptures, which were made of much more valuable materials that could not be wasted by experimentation.