I heard that after injury or repeated damage bone becomes denser, can skin become denser as well?
The structure of skin (epidermis) can change with damage, when the skin is broken the body repairs this partly by recruiting cells called fibroblasts. These fibroblasts lay down collagen fibres to repair the skin. These fibres a normal component of skin are not as neatly arranged as the original fibres and may also higher in number leading to 'denser' skin. This is essentially what a scar is. In some cases the above process can go awry forming what is known as keloid scars.
The other thing that can happen with repeated pressure exposure the skin can become callous, this is when there is thickening of the outer layer formed of cells called keratinocytes increase in number due to recurrent damage. These form typically on pressure points such as balls of feet.