Is it okay to skip "with you" here to keep this sentence have the same meaning?
Yes, it would be okay to drop "with you" in this case and just say "take it on your morning jog" instead. It is okay to do this because the verb "take" in this case already implies that you are going to bring something with you, so you don't need to repeat this here. However, saying "with you" often adds emphasis to the fact that you are bringing something somewhere, and so it is often most common to say "take with you" instead of just "take". Here's an example of "take" and "take with you": Ex: Take these books for the trip. Ex: Take this book with you for the plane. Both sentences mean that someone is bringing something with them, but it is more emphasized with the phrase "take with you".