Consider three (or any number bigger than 2) electrons without spatial degrees of freedom, thus the only degree of freedom is the spins. The Hilbert space is then formed by the tensor product of the space of each electron. Now according to my literature about the so-called antisymmetric tensor, that there are no antisymmetric tensor can be formed if the number of vectors to be tensor-multiplied is bigger than the dimension of each vector space. If applied to my example in the beginning, it seems that I cannot form an antisymmetric state in the system of three electrons because the number of ket to be tensor-multiplied is three (there are three electrons) while the dimension of each is 2 (due to spin 1/2). If this is true, then is three electron system without spatial degrees of freedom impossible? But this is strange.
Yes, it is not possible to construct a totally antisymmetric spin state with more than two electrons. This is just a statement of Pauli's exclusion principle.