Dead skin is something that you have to deal with every day. A year from now, there will be no trace of the skin that you had at the beginning of that year. This is important in a symbolic sense and will afford even more meaning to a dream about dead skin.
If you have a dream of dead skin, this means that you are growing and you are ready to move on from one state in your life. Skin sheds when you have aged or when you have grown. There are also a multitude of random reasons why it might shed, but for purposes of dream interpretation the skin is related back to growth as a human being. If you have a dream where you are having to brush off a lot of dead skin it means that you have grown as a person and are ready to move on from where you used to be, to a new life which you are ready to take on. Don't be afraid of the change; you will grow to love it because it will mean that you have gained a newfound freedom as well.
It means that you have moved on from a point in your life in which you were stagnant. You feel like you need to progress somewhat and this dream is the marker that lets you know you have finally evolved enough to become a different person. Perhaps you have made a great change and this is your brain letting you know that it has been finalized. You really did become a new person.