At first we take two definitions: for the asymmetric metric tensor and for the Christoffel symbols. In order to find the connection of Christoffel symbols with the asymmetric metric tensor we take the derivative from the definition of metric tensor with respect to coordinate. In the resulting equation we make the cyclic rearrangement of indexes two times and get two more equations. Then we rearrange the indexes of metric tensor in these three equations and get three more equations.
After this we represent the metric tensor as the sum of symmetric and antisymmetric tensors. Also we represent the Christoffel symbols as the sum of symmetric and antisymmetric symbols.
Resolving our 6 equations we get the formula for Christoffel symbols expressed through the symmetric and antisymmetric parts of metric tensor.
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