Given two following statements:
$1.$ "The diagonal elements of a skew-symmetric matrix are all zero."
$2.$ "A real/complex square matrix can be uniquely expressed as the sum of a symmetric matrix and a skew-symmetric matrix."
As per my knowledge, these two statements are true. But I have a small doubt, Do these two statements hold every time ? or there are some cases where they don't.
Actually I saw somewhere that "these are not true if the ground field F be of characteristic $2$". But they doesn't provide any further details. So if any one has any point about my doubt, please provide me the same.