This is not gonna be too detailed (though I may work on it in more detail later, if I feel like it).

Class - Race - Primary skills: 1st skill, 2nd skill; Secondary skills: 3rd skill, 4th skill

Official Game Guide Archetypes (may be slightly altered)

Warrior - Nord - 1-Handed, Block; Light Armour/Heavy Armour, Archery

Mage - High Elf (Altmer) - Destruction, Conjuration; Restoration, Illusion

Archer - Wood Elf (Bosmer) - Archery, Sneak; Light Armour, Block

Berserker - Orc (Orsimer) - 2-Handed, Heavy Armour; Smithing, Block

Spellsword Dark Elf (Dunmer) - 1-Handed, Destruction; Light Armour, Illusion

Necromancer - Breton - Conjuration, Illusion; Alteration, Restoration

Assassin - Khajiit - Sneak, 1-Handed (Dual); Alchemy, Pickpocket/Lockpicking

Battlemage - Imperial - Destruction, Restoration; Heavy Armour, Illusion

Weaponmaster - Redguard - 1-Handed (Dual), Heavy Armour; Archery, Smithing/Enchanting

Rogue - Argonian - Sneak, Illusion; Archery, 1-Handed

My Own Archetype Ideas

Knight - Nord/Imperial - 2-Handed, Heavy Armour; Archery, Block

Paladin - Imperial/Orc/Nord - 2-Handed, Restoration; Heavy Armour, Block

Crusader - Redguard/Imperial - 1-Handed (Dual), Heavy Armour; Alteration, Enchanting

Nightblade - Dunmer/Argonian - Sneak, 1-Handed; Destruction, Light Armour

Cleric/Clergyman - RedguardAltmer/Dunmer - Restoration, 1-Handed; Alteration, Destruction

Hunter - Bosmer/Argonian - Sneak, Archery; Light Armour, Alchemy

Bishop - Imperial/Nord - Speech, 2-Handed; Heavy Armour, Restoration

Tradesman - Nord/Imperial - Speech, Alchemy; Archery, Smithing

Enchantress/Enchanter - Breton - Illusion, Enchanting; Speech, Destruction

Sorceror/Sorceress - Altmer/Dunmer - Destruction, Illusion; Alteration, Enchanting/Conjuration

Demonologist - Altmer/Breton - Conjuration, Destruction; Illusion, Heavy Armour

Warlock - Breton/Altmer/Imperial - Destruction, Conjuration; Heavy Armour, Restoration

Reaver - Orc/Dunmer - 1-Handed, Destruction; Enchanting, Heavy Armour

Marksman - Bosmer/Argonian - Archery, Sneak; Alchemy, Light Armour

Thief - Argonian/Bosmer - Pickpocket, Sneak; Lockpick, 1-Handed

Champion - Imperial/Orc/Redguard - 1-Handed, Block; Heavy Armour, Alteration/Speech

Healer - Breton/Imperial/Argonian/Altmer - Restoration, Alchemy; Alteration, Speech

Seal Master/Mistress - Breton/Altmer - Conjuration, Destruction (Runes); Illusion, Restoration

Herbalist - Breton - Alchemy, Restoration; Speech, 1-Handed

If you have any suggestions, or ideas, please comment. I'll be sure to credit you if I use your ideas (not that it'd mean much - unless somehow, lots of people comment on, and read this blog post).