Yama depiction on various Tankas I saw many images of the circle of existence represented with Yama, the lord of death, with a tiger skin and a third eye. Why is he represented in this way?
I believe that it is a depiction of Mahakala, rather than Yama.
In the Chinese Buddhist Encyclopedia page Mahakala is a Protector of all Tibetan Buddhism, we find this info:

The Form varies according to the different teaching lineages. For
example, there is the two-armed, big-mouthed Mahakala Bernakchen of
the Karma Kagyu, the four-armed Mahakala who is protector of the
Drikung Kagyu, and six-armed Mahakala of the Gelugpas described below.
They are not all wrathful forms of Chenresi [Skt.: Avalokiteshvara)
although:

&quot;There is an emanation of Avalokiteshvara which arose in the Form
of Mahakala and this is the Shangpa Kagyu Mahakala with one face and
six arms, in a standing posture. This Form was later adopted by
Tsongkhapa and followers as the main protector of the Gelugpa School.
Mahakalas can arise from various sources namely Vajradhara and
the Anuttarayoga Tantras but are not generically Avalokiteshvara.&quot;
~ Jeff Watt, Himalayan Art

There is a white one associated with prosperity, and also a eunuch and
some feminine forms, but it is incorrectly held that there are 75
variants.

Regarding the symbolism around Mahakala:

Symbolism of Mahakala in the 6-armed Manifestation:
The Protector's Body is midnight blue, symbolic of the changeless
Dharmakaya.
His three eyes symbolize his Knowledge of the past, present and
future, and also the manifestation of the three bodies of Buddha.
The crown adorned with five skulls symbolizes the transformation of
the five poisons of Anger, desire, Ignorance, jealousy and pride into
The Five Wisdoms.
His six arms symbolize the attainment of the six Perfections:
Generosity, Patience, Morality, Diligence, Meditation and Wisdom. The
kartika or triku [or trigu, pron. tigu] the Ritual curved knife, cuts
Attachment to ego.
The kapila or skull bowl filled with blood symbolizes the subjugation
of the maras or Evil ones. (An alternate interpretation can be found
in other contexts.)
The rosary symbolizes his continuous activity for the benefit of
beings.
The Damaru or hand-drum symbolizes his Power over the Dakinis. (Also,
different interpretations in other contexts.)
His trident symbolizes his Power over The Three Kayas -- the spheres
of desire, Form and formlessness. (An alternate interpretation can
also be found.)
The lasso binds those who break their vows.
His two feet are the means and the Wisdom to accomplish his task. That
his left leg is straight and his right leg bent symbolize his
accomplishment of the benefit to oneself and to others. He tramples on
a vinayaka, to symbolize his destruction and dispersal of great
obstacles.
The sun on which he stands symbolizes his illumination of the darkness
of Ignorance.
His Lotus seat symbolizes purity undefiled by Samsara.
The surrounding blazing Fire symbolizes his activity that consumes
neurotic states.
The tiger skin stands for Purification of desire; the elephant skin
for Purification of pride, and the snake, for the Purification of
Anger.
His other ornaments together symbolize that he has all the qualities
of a Buddha.
The material about the symbolism of 6-armed Mahakala derives from a
Gelugpa sadhana (Ritual) of Tara that includes an Offering to
Mahakala.