Why does AlCl3 form the dimer Al2Cl6?

AlCl₃ has a trigonal planar structure. There is an aluminium atom with 3 sp₂ hybridised orbitals which bond to 3 chlorine atoms. The bond angle is 120 degrees. Each chlorine atom shares 1 electron with aluminium in the chemical bond in order to complete it's valence shell. Aluminium has 3 valence electrons and gets 3 more by sharing one with each of the 3 chlorine atoms which brings its total valence electron count to 6 electrons. In order to complete its electron shell aluminium requires 2 more electrons. Chlorine has 6 electrons which aren't involved in bonding which form 3 non bonding pairs. In order to complete aluminium's valence electron shell two AlCl3 molecules dimerise. A chlorine atom from each of the two molecules forms a dative covalent bond with one of its lone electron pairs to the aluminium centre making the molecule Al2Cl6